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# R.D. Sharma solutions for Class 12 Mathematics chapter 7 - Adjoint and Inverse of a Matrix

## Chapter 7 - Adjoint and Inverse of a Matrix

#### Pages 10 - 11

Q 1.1 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}5 & 20 \\ 0 & - 1\end{bmatrix}$

Q 1.2 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}- 1 & 4 \\ 2 & 3\end{bmatrix}$

Q 1.3 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{bmatrix}$

Q 1.4 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{bmatrix}$

Q 1.5 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}0 & 2 & 6 \\ 1 & 5 & 0 \\ 3 & 7 & 1\end{bmatrix}$

Q 1.6 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}a & h & g \\ h & b & f \\ g & f & c\end{bmatrix}$

Q 1.7 | Page 10

Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:

$A = \begin{bmatrix}2 & - 1 & 0 & 1 \\ - 3 & 0 & 1 & - 2 \\ 1 & 1 & - 1 & 1 \\ 2 & - 1 & 5 & 0\end{bmatrix}$

Q 2.1 | Page 10

Evaluate the following determinant:

$\begin{vmatrix}x & - 7 \\ x & 5x + 1\end{vmatrix}$

Q 2.2 | Page 10

Evaluate the following determinant:

$\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}$

Q 2.3 | Page 10

Evaluate the following determinant:

$\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}$

Q 2.4 | Page 10

Evaluate the following determinant:

$\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}$

Q 3 | Page 10

Evaluate

$\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}^2 .$

Q 4 | Page 10

Show that

$\begin{vmatrix}\sin 10^\circ & - \cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ\end{vmatrix} = 1$

Q 5 | Page 10

Evaluate

$\begin{vmatrix}2 & 3 & - 5 \\ 7 & 1 & - 2 \\ - 3 & 4 & 1\end{vmatrix}$ by two methods.

Q 6 | Page 10

Evaluate
$∆ = \begin{vmatrix}0 & \sin \alpha & - \cos \alpha \\ - \sin \alpha & 0 & \sin \beta \\ \cos \alpha & - \sin \beta & 0\end{vmatrix}$

Q 7 | Page 10

$∆ = \begin{vmatrix}\cos \alpha \cos \beta & \cos \alpha \sin \beta & - \sin \alpha \\ - \sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{vmatrix}$

Q 8 | Page 10

If $A = \begin{bmatrix}2 & 5 \\ 2 & 1\end{bmatrix} \text{ and } B = \begin{bmatrix}4 & - 3 \\ 2 & 5\end{bmatrix}$ , verify that |AB| = |A| |B|.

Q 9 | Page 10

If A $\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{bmatrix}$ , then show that |3 A| = 27 |A|.

Q 10.1 | Page 10

Find the value of x, if
$\begin{vmatrix}2 & 4 \\ 5 & 1\end{vmatrix} = \begin{vmatrix}2x & 4 \\ 6 & x\end{vmatrix}$

Q 10.2 | Page 10

Find the value of x, if

$\begin{vmatrix}2 & 3 \\ 4 & 5\end{vmatrix} = \begin{vmatrix}x & 3 \\ 2x & 5\end{vmatrix}$

Q 10.3 | Page 10

Find the value of x, if

$\begin{vmatrix}3 & x \\ x & 1\end{vmatrix} = \begin{vmatrix}3 & 2 \\ 4 & 1\end{vmatrix}$

Q 10.4 | Page 10

Find the value of x, if

$\begin{vmatrix}3x & 7 \\ 2 & 4\end{vmatrix} = 10$ , find the value of x.

Q 10.5 | Page 10

Find the value of x, if

$\begin{vmatrix}x + 1 & x - 1 \\ x - 3 & x + 2\end{vmatrix} = \begin{vmatrix}4 & - 1 \\ 1 & 3\end{vmatrix}$

Q 10.6 | Page 10

Find the value of x, if

$\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & 5 \\ 8 & 3\end{vmatrix}$

Q 11 | Page 11

Find the integral value of x, if $\begin{vmatrix}x^2 & x & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 4\end{vmatrix} = 28 .$

Q 12.1 | Page 11

For what value of x the matrix A is singular?
$A = \begin{bmatrix}1 + x & 7 \\ 3 - x & 8\end{bmatrix}$

Q 12.2 | Page 11

For what value of x the matrix A is singular?

$A = \begin{bmatrix}x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1\end{bmatrix}$

#### Pages 57 - 62

Q 1.1 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38\end{vmatrix}$

Q 1.2 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26\end{vmatrix}$

Q 1.3 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}a & h & g \\ h & b & f \\ g & f & c\end{vmatrix}$

Q 1.4 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{vmatrix}$

Q 1.5 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25\end{vmatrix}$

Q 1.6 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}$

Q 1.7 | Page 57

Evaluate the following determinant:

$\begin{vmatrix}1 & 3 & 9 & 27 \\ 3 & 9 & 27 & 1 \\ 9 & 27 & 1 & 3 \\ 27 & 1 & 3 & 9\end{vmatrix}$

Q 2.01 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{vmatrix}$

Q 2.02 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}$

Q 2.03 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}$

Q 2.04 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}1/a & a^2 & bc \\ 1/b & b^2 & ac \\ 1/c & c^2 & ab\end{vmatrix}$

Q 2.05 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}a + b & 2a + b & 3a + b \\ 2a + b & 3a + b & 4a + b \\ 4a + b & 5a + b & 6a + b\end{vmatrix}$

Q 2.06 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}1 & a & a^2 - bc \\ 1 & b & b^2 - ac \\ 1 & c & c^2 - ab\end{vmatrix}$

Q 2.07 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{vmatrix}$

Q 2.08 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}0 & x & y \\ - x & 0 & z \\ - y & - z & 0\end{vmatrix}$

Q 2.09 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{vmatrix}$

Q 2.1 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}1^2 & 2^2 & 3^2 & 4^2 \\ 2^2 & 3^2 & 4^2 & 5^2 \\ 3^2 & 4^2 & 5^2 & 6^2 \\ 4^2 & 5^2 & 6^2 & 7^2\end{vmatrix}$

Q 2.11 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z\end{vmatrix}$

Q 2.12 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\left( 2^x + 2^{- x} \right)^2 & \left( 2^x - 2^{- x} \right)^2 & 1 \\ \left( 3^x + 3^{- x} \right)^2 & \left( 3^x - 3^{- x} \right)^2 & 1 \\ \left( 4^x + 4^{- x} \right)^2 & \left( 4^x - 4^{- x} \right)^2 & 1\end{vmatrix}$

Q 2.13 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\sin\alpha & \cos\alpha & \cos(\alpha + \delta) \\ \sin\beta & \cos\beta & \cos(\beta + \delta) \\ \sin\gamma & \cos\gamma & \cos(\gamma + \delta)\end{vmatrix}$

Q 2.14 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\sin^2 23^\circ & \sin^2 67^\circ & \cos180^\circ \\ - \sin^2 67^\circ & - \sin^2 23^\circ & \cos^2 180^\circ \\ \cos180^\circ & \sin^2 23^\circ & \sin^2 67^\circ\end{vmatrix}$

Q 2.15 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}$

Q 2.16 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5\end{vmatrix}$

Q 2.17 | Page 57

Without expanding, show that the value of the following determinant is zero:

$\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}, where A, B, C \text{ are the angles of }∆ ABC .$

Q 3 | Page 58

Evaluate :

$\begin{vmatrix}a & b + c & a^2 \\ b & c + a & b^2 \\ c & a + b & c^2\end{vmatrix}$

Q 4 | Page 58

Evaluate :

$\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix}$

Q 5 | Page 58

Evaluate :

$\begin{vmatrix}x + \lambda & x & x \\ x & x + \lambda & x \\ x & x & x + \lambda\end{vmatrix}$

Q 6 | Page 58

Evaluate :

$\begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix}$

Q 7 | Page 58

Evaluate the following:

$\begin{vmatrix}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{vmatrix}$

Q 8 | Page 58

Evaluate the following:

$\begin{vmatrix}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{vmatrix}$

Q 9 | Page 58

Evaluate the following:

$\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}$

Q 10 | Page 58

$If ∆ = \begin{vmatrix}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{vmatrix}, ∆_1 = \begin{vmatrix}1 & 1 & 1 \\ yz & zx & xy \\ x & y & z\end{vmatrix},\text{ then prove that }∆ + ∆_1 = 0 .$

Q 11 | Page 58

Prove that :

$\begin{vmatrix}a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b\end{vmatrix} = a^3 + b^3 + c^3 - 3abc$

Q 12 | Page 58

Prove that :

$\begin{vmatrix}b + c & a - b & a \\ c + a & b - c & b \\ a + b & c - a & c\end{vmatrix} = 3abc - a^3 - b - c^3$

Q 13 | Page 58

Prove that :

$\begin{vmatrix}a + b & b + c & c + a \\ b + c & c + a & a + b \\ c + a & a + b & b + c\end{vmatrix} = 2\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}$

Q 14 | Page 58

Prove that :

$\begin{vmatrix}a + b + 2c & a & b \\ c & b + c + 2a & b \\ c & a & c + a + 2b\end{vmatrix} = 2 \left( a + b + c \right)^3$

Q 15 | Page 59

Prove that :

$\begin{vmatrix}a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b\end{vmatrix} = \left( a + b + c \right)^3$

Q 16 | Page 59

Prove that :

$\begin{vmatrix}1 & b + c & b^2 + c^2 \\ 1 & c + a & c^2 + a^2 \\ 1 & a + b & a^2 + b^2\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right)$

Q 17 | Page 59

Prove that :

$\begin{vmatrix}a & a + b & a + 2b \\ a + 2b & a & a + b \\ a + b & a + 2b & a\end{vmatrix} = 9 \left( a + b \right) b^2$

Q 18 | Page 59

Prove that :

$\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix} = \begin{vmatrix}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{vmatrix}$

Q 19 | Page 59

Prove that :

$\begin{vmatrix}z & x & y \\ z^2 & x^2 & y^2 \\ z^4 & x^4 & y^4\end{vmatrix} = \begin{vmatrix}x & y & z \\ x^2 & y^2 & z^2 \\ x^4 & y^4 & z^4\end{vmatrix} = \begin{vmatrix}x^2 & y^2 & z^2 \\ x^4 & y^4 & z^4 \\ x & y & z\end{vmatrix} = xyz \left( x - y \right) \left( y - z \right) \left( z - x \right) \left( x + y + z \right) .$

Q 20 | Page 59

Prove that :

$\begin{vmatrix}\left( b + c \right)^2 & a^2 & bc \\ \left( c + a \right)^2 & b^2 & ca \\ \left( a + b \right)^2 & c^2 & ab\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a + b + c \right) \left( a^2 + b^2 + c^2 \right)$

Q 21 | Page 59

Prove that :

$\begin{vmatrix}\left( a + 1 \right) \left( a + 2 \right) & a + 2 & 1 \\ \left( a + 2 \right) \left( a + 3 \right) & a + 3 & 1 \\ \left( a + 3 \right) \left( a + 4 \right) & a + 4 & 1\end{vmatrix} = - 2$

Q 22 | Page 59

Prove that :

$\begin{vmatrix}a^2 & a^2 - \left( b - c \right)^2 & bc \\ b^2 & b^2 - \left( c - a \right)^2 & ca \\ c^2 & c^2 - \left( a - b \right)^2 & ab\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a + b + c \right) \left( a^2 + b^2 + c^2 \right)$

Q 23 | Page 59

Prove that :

$\begin{vmatrix}1 & a^2 + bc & a^3 \\ 1 & b^2 + ca & b^3 \\ 1 & c^2 + ab & c^3\end{vmatrix} = - \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a^2 + b^2 + c^2 \right)$

Q 24 | Page 59

Prove that :

$\begin{vmatrix}a^2 & bc & ac + c^2 \\ a^2 + ab & b^2 & ac \\ ab & b^2 + bc & c^2\end{vmatrix} = 4 a^2 b^2 c^2$
Q 25 | Page 59

Prove that :

$\begin{vmatrix}x + 4 & x & x \\ x & x + 4 & x \\ x & x & x + 4\end{vmatrix} = 16 \left( 3x + 4 \right)$
Q 26 | Page 59

Prove that :

$\begin{vmatrix}1 & 1 + p & 1 + p + q \\ 2 & 3 + 2p & 4 + 3p + 2q \\ 3 & 6 + 3p & 10 + 6p + 3q\end{vmatrix} = 1$

Q 27 | Page 59

Prove that :

$\begin{vmatrix}a & b - c & c - b \\ a - c & b & c - a \\ a - b & b - a & c\end{vmatrix} = \left( a + b - c \right) \left( b + c - a \right) \left( c + a - b \right)$

Q 28 | Page 60

Prove that

$\begin{vmatrix}a^2 & 2ab & b^2 \\ b^2 & a^2 & 2ab \\ 2ab & b^2 & a^2\end{vmatrix} = \left( a^3 + b^3 \right)^2$
Q 29 | Page 60

Prove that

$\begin{vmatrix}a^2 + 1 & ab & ac \\ ab & b^2 + 1 & bc \\ ca & cb & c^2 + 1\end{vmatrix} = 1 + a^2 + b^2 + c^2$
Q 30 | Page 60
$\begin{vmatrix}1 & a & a^2 \\ a^2 & 1 & a \\ a & a^2 & 1\end{vmatrix} = \left( a^3 - 1 \right)^2$
Q 31 | Page 60
$\begin{vmatrix}a + b + c & - c & - b \\ - c & a + b + c & - a \\ - b & - a & a + b + c\end{vmatrix} = 2\left( a + b \right) \left( b + c \right) \left( c + a \right)$
Q 32 | Page 60

$\begin{vmatrix}b + c & a & a \\ b & c + a & b \\ c & c & a + b\end{vmatrix} = 4abc$

Q 33 | Page 60

$\begin{vmatrix}b^2 + c^2 & ab & ac \\ ba & c^2 + a^2 & bc \\ ca & cb & a^2 + b^2\end{vmatrix} = 4 a^2 b^2 c^2$

Q 34 | Page 60

$\begin{vmatrix}0 & b^2 a & c^2 a \\ a^2 b & 0 & c^2 b \\ a^2 c & b^2 c & 0\end{vmatrix} = 2 a^3 b^3 c^3$

Q 35 | Page 60

Prove that

$\begin{vmatrix}\frac{a^2 + b^2}{c} & c & c \\ a & \frac{b^2 + c^2}{a} & a \\ b & b & \frac{c^2 + a^2}{b}\end{vmatrix} = 4abc$

Q 36 | Page 58

Prove that
$\begin{vmatrix}- bc & b^2 + bc & c^2 + bc \\ a^2 + ac & - ac & c^2 + ac \\ a^2 + ab & b^2 + ab & - ab\end{vmatrix} = \left( ab + bc + ca \right)^3$

Q 37 | Page 60

Prove the following identities:
$\begin{vmatrix}x + \lambda & 2x & 2x \\ 2x & x + \lambda & 2x \\ 2x & 2x & x + \lambda\end{vmatrix} = \left( 5x + \lambda \right) \left( \lambda - x \right)^2$

Q 38 | Page 60

Using properties of determinants prove that

$\begin{vmatrix}x + 4 & 2x & 2x \\ 2x & x + 4 & 2x \\ 2x & 2x & x + 4\end{vmatrix} = \left( 5x + 4 \right) \left( 4 - x \right)^2$

Q 39 | Page 60

Prove the following identities:

$\begin{vmatrix}y + z & z & y \\ z & z + x & x \\ y & x & x + y\end{vmatrix} = 4xyz$

Q 40 | Page 61

$\begin{vmatrix}- a \left( b^2 + c^2 - a^2 \right) & 2 b^3 & 2 c^3 \\ 2 a^3 & - b \left( c^2 + a^2 - b^2 \right) & 2 c^3 \\ 2 a^3 & 2 b^3 & - c \left( a^2 + b^2 - c^2 \right)\end{vmatrix} = abc \left( a^2 + b^2 + c^2 \right)^3$

Q 41 | Page 61

$\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + a & a \\ 1 & 1 & 1 + a\end{vmatrix} = a^3 + 3 a^2$

Q 42 | Page 61

Prove the following identity:

$\begin{vmatrix}2y & y - z - x & 2y \\ 2z & 2z & z - x - y \\ x - y - z & 2x & 2x\end{vmatrix} = \left( x + y + z \right)^3$

Q 44 | Page 61

Prove the following identity:

$\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)$

Q 45 | Page 61

Prove the following identity:

|(a^3,2,a),(b^3,2,b),(c^3,2,c)| = 2(a-b) (b-c) (c-a) (a+b+c)

Q 46 | Page 61

Without expanding, prove that

$\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix} = \begin{vmatrix}x & y & z \\ p & q & r \\ a & b & c\end{vmatrix} = \begin{vmatrix}y & b & q \\ x & a & p \\ z & c & r\end{vmatrix}$

Q 47 | Page 61

Show that

$\begin{vmatrix}x + 1 & x + 2 & x + a \\ x + 2 & x + 3 & x + b \\ x + 3 & x + 4 & x + c\end{vmatrix} =\text{ 0 where a, b, c are in A . P .}$

Q 48 | Page 61
Show that
|(x-3,x-4,x-alpha),(x-2,x-3,x-beta),(x-1,x-2,x-gamma)|=0, where α, β, γ are in A.P.

Q 49 | Page 61

If a, b, c are real numbers such that
$\begin{vmatrix}b + c & c + a & a + b \\ c + a & a + b & b + c \\ a + b & b + c & c + a\end{vmatrix} = 0$ , then show that either
$a + b + c = 0 \text{ or, } a = b = c$

Q 50 | Page 61
$If \begin{vmatrix}p & b & c \\ a & q & c \\ a & b & r\end{vmatrix} = 0,\text{ find the value of }\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c}, p \neq a, q \neq b, r \neq c .$

Q 51 | Page 61

Show that x = 2 is a root of the equation

$\begin{vmatrix}x & - 6 & - 1 \\ 2 & - 3x & x - 3 \\ - 3 & 2x & x + 2\end{vmatrix} = 0$  and solve it completely.

Q 52.1 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}x + a & b & c \\ a & x + b & c \\ a & b & x + c\end{vmatrix} = 0$

Q 52.2 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}x + a & x & x \\ x & x + a & x \\ x & x & x + a\end{vmatrix} = 0, a \neq 0$

Q 52.3 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8\end{vmatrix} = 0$

Q 52.4 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}1 & x & x^2 \\ 1 & a & a^2 \\ 1 & b & b^2\end{vmatrix} = 0, a \neq b$

Q 52.5 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4\end{vmatrix} = 0$

Q 52.6 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}1 & x & x^3 \\ 1 & b & b^3 \\ 1 & c & c^3\end{vmatrix} = 0, b \neq c$

Q 52.7 | Page 61
​Solve the following determinant equation:
$\begin{vmatrix}15 - 2x & 11 - 3x & 7 - x \\ 11 & 17 & 14 \\ 10 & 16 & 13\end{vmatrix} = 0$
Q 52.8 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}1 & 1 & x \\ p + 1 & p + 1 & p + x \\ 3 & x + 1 & x + 2\end{vmatrix} = 0$
Q 52.9 | Page 61

​Solve the following determinant equation:

$\begin{vmatrix}3 & - 2 & \sin\left( 3\theta \right) \\ - 7 & 8 & \cos\left( 2\theta \right) \\ - 11 & 14 & 2\end{vmatrix} = 0$

Q 53 | Page 62

If $a, b$ and c  are all non-zero and

$\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c\end{vmatrix} =$ 0, then prove that
$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} +$1
= 0

Q 54 | Page 62

If $\begin{vmatrix}a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} =$ 0, then using properties of determinants, find the value of  $\frac{a}{x} + \frac{b}{y} + \frac{c}{z}$  , where $x, y, z \neq$ 0

#### Pages 71 - 72

Q 1.1 | Page 71

Find the area of the triangle with vertice at the point:

(3, 8), (−4, 2) and (5, −1)

Q 1.2 | Page 71

Find the area of the triangle with vertice at the point:

(2, 7), (1, 1) and (10, 8)

Q 1.3 | Page 71

Find the area of the triangle with vertice at the point:

(−1, −8), (−2, −3) and (3, 2)

Q 1.4 | Page 71

Find the area of the triangle with vertice at the point:

(0, 0), (6, 0) and (4, 3)

Q 2.1 | Page 71

Using determinants show that the following points are collinear:

(5, 5), (−5, 1) and (10, 7)

Q 2.2 | Page 71

Using determinants show that the following points are collinear:

(1, −1), (2, 1) and (4, 5)

Q 2.3 | Page 71

Using determinants show that the following points are collinear:

(3, −2), (8, 8) and (5, 2)

Q 2.4 | Page 71

Using determinants show that the following points are collinear:

(2, 3), (−1, −2) and (5, 8)

Q 3 | Page 71

If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.

Q 4 | Page 71

Using determinants prove that the points (ab), (a', b') and (a − a', b − b') are collinear if ab' = a'b.

Q 5 | Page 71

Find the value of $\lambda$  so that the points (1, −5), (−4, 5) and $\lambda$  are collinear.

Q 6 | Page 71

Find the value of x if the area of ∆ is 35 square cms with vertices (x, 4), (2, −6) and (5, 4).

Q 7 | Page 71

Using determinants, find the area of the triangle whose vertices are (1, 4), (2, 3) and (−5, −3). Are the given points collinear?

Q 8 | Page 71

Using determinants, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).

Q 9 | Page 71

Using determinants, find the value of k so that the points (k, 2 − 2 k), (−k + 1, 2k) and (−4 − k, 6 − 2k) may be collinear.

Q 10 | Page 71

If the points (x, −2), (5, 2), (8, 8) are collinear, find x using determinants.

Q 11 | Page 72

If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.

Q 12.1 | Page 72

Using determinants, find the equation of the line joining the points

(1, 2) and (3, 6)

Q 12.2 | Page 72

Using determinants, find the equation of the line joining the points

(3, 1) and (9, 3)

Q 13.1 | Page 72

Find values of k, if area of triangle is 4 square units whose vertices are
(k, 0), (4, 0), (0, 2)

Q 13.2 | Page 72

Find values of k, if area of triangle is 4 square units whose vertices are

(−2, 0), (0, 4), (0, k)

#### Pages 84 - 85

Q 1 | Page 84

x − 2y = 4
−3x + 5y = −7

Q 2 | Page 84

2x − y = 1
7x − 2y = −7

Q 3 | Page 84

2x − y = 17
3x + 5y = 6

Q 4 | Page 84

3x + y = 19
3x − y = 23

Q 5 | Page 84

2x − y = − 2
3x + 4y = 3

Q 6 | Page 84

3x + ay = 4
2x + ay = 2, a ≠ 0

Q 7 | Page 84

2x + 3y = 10
x + 6y = 4

Q 8 | Page 84

5x + 7y = − 2
4x + 6y = − 3

Q 9 | Page 84

9x + 5y = 10
3y − 2x = 8

Q 10 | Page 84

Given: x + 2y = 1
3x + y = 4

Q 11 | Page 84

3x + y + z = 2
2x − 4y + 3z = − 1
4x + y − 3z = − 11

Q 12 | Page 84

x − 4y − z = 11
2x − 5y + 2z = 39
− 3x + 2y + z = 1

Q 13 | Page 84

6x + y − 3z = 5
x + 3y − 2z = 5
2x + y + 4z = 8

Q 14 | Page 84

xy = 5
y + z = 3
x + z = 4

Q 15 | Page 84

2y − 3z = 0
x + 3y = − 4
3x + 4y = 3

Q 17 | Page 84

5x − 7y + z = 11
6x − 8y − z = 15
3x + 2y − 6z = 7

Q 18 | Page 84

2x − 3y − 4z = 29
− 2x + 5y − z = − 15
3x − y + 5z = − 11

Q 19 | Page 84

x + y + z + 1 = 0
ax + by + cz + d = 0
a2x + b2y + x2z + d2 = 0

Q 20 | Page 84

x + y + z + w = 2
x − 2y + 2z + 2w = − 6
2x + y − 2z + 2w = − 5
3x − y + 3z − 3w = − 3

Q 21 | Page 84

2x − 3z + w = 1
x − y + 2w = 1
− 3y + z + w = 1
x + y + z = 1

Q 22 | Page 84

2x − y = 5
4x − 2y = 7

Q 23 | Page 84

3x + y = 5
− 6x − 2y = 9

Q 24 | Page 84

3x − y + 2z = 3
2x + y + 3z = 5
x − 2y − z = 1

Q 25 | Page 84

3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.

Q 26 | Page 85

x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1

Q 27 | Page 85

x + 2y = 5
3x + 6y = 15

Q 28 | Page 85

x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0

Q 29 | Page 85

2x + y − 2z = 4
x − 2y + z = − 2
5x − 5y + z = − 2

Q 30 | Page 85

x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10

Q 31 | Page 85

A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission

 Month Sale of units Total commissiondrawn (in Rs) A B C Jan 90 100 20 800 Feb 130 50 40 900 March 60 100 30 850

Find out the rates of commission on items A, B and C by using determinant method.

Q 32 | Page 85

An automobile company uses three types of steel S1S2 and S3 for producing three types of cars C1C2and C3. Steel requirements (in tons) for each type of cars are given below :

 CarsC1 C2 C3 Steel S1 2 3 4 S2 1 1 2 S3 3 2 1

Using Cramer's rule, find the number of cars of each type which can be produced using 29, 13 and 16 tons of steel of three types respectively.

#### Page 89

Q 1 | Page 89

Solve each of the following system of homogeneous linear equations.
x + y − 2z = 0
2x + y − 3z = 0
5x + 4y − 9z = 0

Q 2 | Page 89

Solve each of the following system of homogeneous linear equations.
2x + 3y + 4z = 0
x + y + z = 0
2x − y + 3z = 0

Q 3 | Page 89

Solve each of the following system of homogeneous linear equations.
3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0

Q 4 | Page 89

Find the real values of λ for which the following system of linear equations has non-trivial solutions. Also, find the non-trivial solutions
$2 \lambda x - 2y + 3z = 0$
$x + \lambda y + 2z = 0$
$2x + \lambda z = 0$

Q 5 | Page 89

If a, b, c are non-zero real numbers and if the system of equations
(a − 1) x = y + z
(b − 1) y = z + x
(c − 1) z = x + y
has a non-trivial solution, then prove that ab + bc + ca = abc.

#### Pages 90 - 95

Q 1 | Page 90

If A is a singular matrix, then write the value of |A|.

Q 2 | Page 90

For what value of x, the following matrix is singular?

$\begin{bmatrix}5 - x & x + 1 \\ 2 & 4\end{bmatrix}$

Q 3 | Page 90

Write the value of the determinant
$\begin{bmatrix}2 & 3 & 4 \\ 2x & 3x & 4x \\ 5 & 6 & 8\end{bmatrix} .$

Q 4 | Page 90

State whether the matrix
$\begin{bmatrix}2 & 3 \\ 6 & 4\end{bmatrix}$ is singular or non-singular.

Q 5 | Page 90

Find the value of the determinant
$\begin{bmatrix}4200 & 4201 \\ 4205 & 4203\end{bmatrix}$

Q 6 | Page 90

Find the value of the determinant
$\begin{bmatrix}101 & 102 & 103 \\ 104 & 105 & 106 \\ 107 & 108 & 109\end{bmatrix}$

Q 7 | Page 90

Write the value of the determinant

$\begin{vmatrix}a & 1 & b + c \\ b & 1 & c + a \\ c & 1 & a + b\end{vmatrix} .$

Q 8 | Page 90

If $A = \begin{bmatrix}0 & i \\ i & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$ , find the value of |A| + |B|.

Q 9 | Page 90

If $A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ - 1 & 0\end{bmatrix}$ , find |AB|.

Q 10 | Page 90

Evaluate $\begin{vmatrix}4785 & 4787 \\ 4789 & 4791\end{vmatrix}$

Q 11 | Page 90

If w is an imaginary cube root of unity, find the value of $\begin{vmatrix}1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w\end{vmatrix}$

Q 12 | Page 90

If $A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and B} = \begin{bmatrix}1 & - 4 \\ 3 & - 2\end{bmatrix},\text{ find }|AB|$

Q 13 | Page 90

If $A = \left[ a_{ij} \right]$   is a 3 × 3 diagonal matrix such that a11 = 1, a22 = 2 a33 = 3, then find |A|.

Q 14 | Page 90

If A = [aij] is a 3 × 3 scalar matrix such that a11 = 2, then write the value of |A|.

Q 15 | Page 90

If I3 denotes identity matrix of order 3 × 3, write the value of its determinant.

Q 16 | Page 90

A matrix A of order 3 × 3 has determinant 5. What is the value of |3A|?

Q 17 | Page 90

On expanding by first row, the value of the determinant of 3 × 3 square matrix
$A = \left[ a_{ij} \right]\text{ is }a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}$ , where [Cij] is the cofactor of aij in A. Write the expression for its value on expanding by second column.

Q 18 | Page 90

Let A = [aij] be a square matrix of order 3 × 3 and Cij denote cofactor of aij in A. If |A| = 5, write the value of a31 C31  +  a32 C32 a33 C33.

Q 19 | Page 90

In question 18, write the value of a11 C21 + a12 C22 + a13 C23.

Q 20 | Page 90

Write the value of

$\begin{vmatrix}\sin 20^\circ & - \cos 20^\circ\\ \sin 70^\circ& \cos 70^\circ\end{vmatrix}$
Q 21 | Page 90

If A is a square matrix satisfying AT A = I, write the value of |A|.

Q 22 | Page 91

If A and B are square matrices of the same order such that |A| = 3 and AB = I, then write the value of |B|.

Q 23 | Page 91

A is a skew-symmetric of order 3, write the value of |A|.

Q 24 | Page 91

If A is a square matrix of order 3 with determinant 4, then write the value of |−A|.

Q 25 | Page 91

If A is a square matrix such that |A| = 2, write the value of |A AT|.

Q 26 | Page 91

Find the value of the determinant $\begin{vmatrix}243 & 156 & 300 \\ 81 & 52 & 100 \\ - 3 & 0 & 4\end{vmatrix} .$

Q 27 | Page 91

Write the value of the determinant $\begin{vmatrix}2 & - 3 & 5 \\ 4 & - 6 & 10 \\ 6 & - 9 & 15\end{vmatrix} .$

Q 28 | Page 91

If the matrix $\begin{bmatrix}5x & 2 \\ - 10 & 1\end{bmatrix}$  is singular, find the value of x.

Q 29 | Page 91

If A is a square matrix of order n × n such that  $|A| = \lambda$ , then write the value of |−A|.

Q 30 | Page 91

Find the value of the determinant $\begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix}$.

Q 31 | Page 91

If A and B are non-singular matrices of the same order, write whether AB is singular or non-singular.

Q 32 | Page 91

A matrix of order 3 × 3 has determinant 2. What is the value of |A (3I)|, where I is the identity matrix of order 3 × 3.

Q 33 | Page 91

If A and B are square matrices of order 3 such that |A| = − 1, |B| = 3, then find the value of |3 AB|.

Q 34 | Page 91

Write the value of  $\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix} .$

Q 35 | Page 91

Write the cofactor of a12 in the following matrix $\begin{bmatrix}2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7\end{bmatrix} .$

Q 36 | Page 91

If $\begin{vmatrix}2x + 5 & 3 \\ 5x + 2 & 9\end{vmatrix} = 0$

Q 37 | Page 91

Find the value of x from the following : $\begin{vmatrix}x & 4 \\ 2 & 2x\end{vmatrix} = 0$

Q 38 | Page 91

Write the value of the determinant $\begin{vmatrix}2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x\end{vmatrix}$

Q 39 | Page 91

If |A| = 2, where A is 2 × 2 matrix, find |adj A|.

Q 40 | Page 91

What is the value of the determinant $\begin{vmatrix}0 & 2 & 0 \\ 2 & 3 & 4 \\ 4 & 5 & 6\end{vmatrix} ?$

Q 41 | Page 91

For what value of x is the matrix  $\begin{bmatrix}6 - x & 4 \\ 3 - x & 1\end{bmatrix}$  singular?

Q 42 | Page 91

A matrix A of order 3 × 3 is such that |A| = 4. Find the value of |2 A|.

Q 43 | Page 92

Evaluate: $\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}$

Q 44 | Page 92

If $A = \begin{bmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{bmatrix}$. Write the cofactor of the element a32.

Q 45 | Page 92
If $\begin{vmatrix}x + 1 & x - 1 \\ x - 3 & x + 2\end{vmatrix} = \begin{vmatrix}4 & - 1 \\ 1 & 3\end{vmatrix}$, then write the value of x.
Q 46 | Page 92

If $\begin{vmatrix}2x & x + 3 \\ 2\left( x + 1 \right) & x + 1\end{vmatrix} = \begin{vmatrix}1 & 5 \\ 3 & 3\end{vmatrix}$, then write the value of x.

Q 47 | Page 92

If $\begin{vmatrix}3x & 7 \\ - 2 & 4\end{vmatrix} = \begin{vmatrix}8 & 7 \\ 6 & 4\end{vmatrix}$ , find the value of x.

Q 48 | Page 92

If $\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}$ , write the value of x.

Q 49 | Page 92

If A is a 3 × 3 matrix, $\left| A \right| \neq 0\text{ and }\left| 3A \right| = k\left| A \right|$  then write the value of k.

Q 50 | Page 92

Write the value of the determinant $\begin{vmatrix}p & p + 1 \\ p - 1 & p\end{vmatrix}$

Q 51 | Page 92

Write the value of the determinant $\begin{vmatrix}x + y & y + z & z + x \\ z & x & y \\ - 3 & - 3 & - 3\end{vmatrix}$

Q 52 | Page 92

If $A = \begin{bmatrix}\cos\theta & \sin\theta \\ - \sin\theta & \cos\theta\end{bmatrix}$ , then for any natural number, find the value of Det(An).

Q 53 | Page 95

Find the maximum value of $\begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 & 1 & 1 + \cos \theta\end{vmatrix}$

Q 54 | Page 95

If x ∈ N and $\begin{vmatrix}x + 3 & - 2 \\ - 3x & 2x\end{vmatrix}$  = 8, then find the value of x.

Q 55 | Page 95

If $\begin{vmatrix}x & \sin \theta & \cos \theta \\ - \sin \theta & - x & 1 \\ \cos \theta & 1 & x\end{vmatrix} = 8$ , write the value of x.

Q 56 | Page 95

If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A–1) = (det A)k.

#### Pages 93 - 96

Q 1 | Page 93

If A and B are square matrices of order 2, then det (A + B) = 0 is possible only when
(a) det (A) = 0 or det (B) = 0
(b) det (A) + det (B) = 0
(c) det (A) = 0 and det (B) = 0
(d) A + B = O

Q 2 | Page 93

Which of the following is not correct?
(a) $|A| = | A^T |,\text{ where }A = \left[ a_{ij} \right]_{3 \times 3}$
(b) $|kA| = | k^3 |,\text{ where }A = \left[ a_{ij} \right]_{3 \times 3}$
(c) If A is a skew-symmetric matrix of odd order, then |A| = 0
(d) $\begin{vmatrix}a + b & c + d \\ e + f & g + h\end{vmatrix} = \begin{vmatrix}a & c \\ e & g\end{vmatrix} + \begin{vmatrix}b & d \\ f & h\end{vmatrix}$

Q 3 | Page 93

If $A = \begin{vmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{vmatrix}$  and Cij is cofactor of aij in A, then value of |A| is given
(a) a11 C31 + a12 C32 + a13 C33
(b) a11 C11 + a12 C21 + a13 C31
(c) a21 C11 + a22 C12 + a23 C13
(d) a11 C11 + a21 C21 + a13 C31

Q 4 | Page 93

Which of the following is not correct in a given determinant of A, where A = [aij]3×3.
(a) Order of minor is less than order of the det (A)
(b) Minor of an element can never be equal to cofactor of the same element
(c) Value of determinant is obtained by multiplying elements of a row or column by       corresponding cofactors
(d) Order of minors and cofactors of elements of A is same

Q 5 | Page 93

Let $\begin{vmatrix}x & 2 & x \\ x^2 & x & 6 \\ x & x & 6\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e$
Then, the value of $5a + 4b + 3c + 2d + e$ is equal to
(a) 0
(b) − 16
(c) 16
(d) none of these

Q 6 | Page 93

The value of the determinant

$\begin{vmatrix}a^2 & a & 1 \\ \cos nx & \cos \left( n + 1 \right) x & \cos \left( n + 2 \right) x \\ \sin nx & \sin \left( n + 1 \right) x & \sin \left( n + 2 \right) x\end{vmatrix}\text{ is independent of}$
(a) n
(b) a
(c) x
(d) none of these

Q 7 | Page 93

If  $∆_1 = \begin{vmatrix}1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2\end{vmatrix}, ∆_2 = \begin{vmatrix}1 & bc & a \\ 1 & ca & b \\ 1 & ab & c\end{vmatrix},\text{ then }$}
(a) $∆_1 + ∆_2 = 0$
(b) $∆_1 + 2 ∆_2 = 0$
(c) $∆_1 = ∆_2$
(d) none of these

Q 8 | Page 93
If $D_k = \begin{vmatrix}1 & n & n \\ 2k & n^2 + n + 2 & n^2 + n \\ 2k - 1 & n^2 & n^2 + n + 2\end{vmatrix} and \sum^n_{k = 1} D_k = 48$, then n equals

(a) 4
(b) 6
(c) 8
(d) none of these

Q 9 | Page 93

Let $\begin{vmatrix}x^2 + 3x & x - 1 & x + 3 \\ x + 1 & - 2x & x - 4 \\ x - 3 & x + 4 & 3x\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e$
be an identity in x, where abcde are independent of x. Then the value of e is
(a) 4
(b) 0
(c) 1
(d) none of these

Q 10 | Page 94

Using the factor theorem it is found that a + bb + c and c + a are three factors of the determinant

$\begin{vmatrix}- 2a & a + b & a + c \\ b + a & - 2b & b + c \\ c + a & c + b & - 2c\end{vmatrix}$
The other factor in the value of the determinant is
(a) 4
(b) 2
(c) a + b + c
(d) none of these
Q 11 | Page 94

If a, b, c are distinct, then the value of x satisfying $\begin{vmatrix}0 & x^2 - a & x^3 - b \\ x^2 + a & 0 & x^2 + c \\ x^4 + b & x - c & 0\end{vmatrix} = 0\text{ is }$
(a) c
(b) a
(c) b
(d) 0

Q 12 | Page 94

If the determinant $\begin{vmatrix}0 & x^2 - a & x^3 - b \\ x^2 + a & 0 & x^2 + c \\ x^4 + b & x - c & 0\end{vmatrix} = 0 \text{ is }$
(a) a, b, c are in H . P $(b) \alpha\text{ is a root of 4a} x^2 + 12bx + 9c = 0\text{ or a, b, c are in G . P .}$
(c) a, b, c are in G . P . only
(d) a, b, c are in A . P .

Q 13 | Page 94

If ω is a non-real cube root of unity and n is not a multiple of 3, then  $∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}$
(a) 0
(b) ω
(c) ω2
(d) 1

Q 14 | Page 94

If $A_r = \begin{vmatrix}1 & r & 2^r \\ 2 & n & n^2 \\ n & \frac{n \left( n + 1 \right)}{2} & 2^{n + 1}\end{vmatrix}$ , then the value of $\sum^n_{r = 1} A_r$ is
(a) n
(b) 2n
(c) − 2n
(d) n2

Q 15 | Page 94

If a > 0 and discriminant of ax2 + 2bx + c is negative, then
$∆ = \begin{vmatrix}a & b & ax + b \\ b & c & bx + c \\ ax + b & bx + c & 0\end{vmatrix} is$

(a) positive
(b) $\left( ac - b^2 \right) \left( a x^2 + 2bx + c \right)$
(c) negative
(d) 0

Q 16 | Page 94

The value of $\begin{vmatrix}5^2 & 5^3 & 5^4 \\ 5^3 & 5^4 & 5^5 \\ 5^4 & 5^5 & 5^6\end{vmatrix}$
(a) 52
(b) 0
(c) 513
(d) 59

Q 17 | Page 94
$\begin{vmatrix}\log_3 512 & \log_4 3 \\ \log_3 8 & \log_4 9\end{vmatrix} \times \begin{vmatrix}\log_2 3 & \log_8 3 \\ \log_3 4 & \log_3 4\end{vmatrix}$
(a) 7
(b) 10
(c) 1
(d) 17
Q 18 | Page 94

If a, b, c are in A.P., then the determinant
$\begin{vmatrix}x + 2 & x + 3 & x + 2a \\ x + 3 & x + 4 & x + 2b \\ x + 4 & x + 5 & x + 2c\end{vmatrix}$
(a) 0
(b) 1
(c) x
(d) 2x

Q 19 | Page 95

If $A + B + C = \pi$, then the value of $\begin{vmatrix}\sin \left( A + B + C \right) & \sin \left( A + C \right) & \cos C \\ - \sin B & 0 & \tan A \\ \cos \left( A + B \right) & \tan \left( B + C \right) & 0\end{vmatrix}$  is equal to
(a) 0
(b) 1
(c) 2 sin B tan A cos C
(d) none of these

Q 20 | Page 95

The number of distinct real roots of $\begin{vmatrix}cosec x & \sec x & \sec x \\ \sec x & cosec x & \sec x \\ \sec x & \sec x & cosec x\end{vmatrix} = 0$  lies in the interval
$- \frac{\pi}{4} \leq x \leq \frac{\pi}{4}$
(a) 1
(b) 2
(c) 3
(d) 0

Q 21 | Page 95

Let $A = \begin{bmatrix}1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1\end{bmatrix},\text{ where 0 }\leq \theta \leq 2\pi . \text{ Then,}$
(a) $Det \left( A \right) = 0$
(b) $Det \left( A \right) \in \left( 2, \infty \right)$
(c) $Det \left( A \right) \in \left( 2, 4 \right)$
(d) $Det \left( A \right) \in \left[ 2, 4 \right]$

Q 22 | Page 95

If $\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}$
(a) 3
(b) ± 3
(c) ± 6
(d) 6

Q 23 | Page 95
If$f\left( x \right) = \begin{vmatrix}0 & x - a & x - b \\ x + a & 0 & x - c \\ x + b & x + c & 0\end{vmatrix}$
(a) f(a) = 0
(b) f(b) = 0
(c) f(0) = 0
(d) f(1) = 0
Q 24 | Page 95

The value of the determinant

$\begin{vmatrix}a - b & b + c & a \\ b - c & c + a & b \\ c - a & a + b & c\end{vmatrix}$
(a)  $a^3 + b^3 + c^3$
(b) 3bc
(c) $a^3 + b^3 + c^3 - 3abc$
(d) none of these
Q 25 | Page 95

If xyare different from zero and $\begin{vmatrix}1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z\end{vmatrix} = 0$ , then the value of x−1 + y−1 + z−1 is
(a) xyz
(b) x−1 y−1 z−1
(c) − x − y − z
(d) − 1

Q 26 | Page 95

The determinant  $\begin{vmatrix}b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 - ab \\ bc - ca & c - a & ab - a^2\end{vmatrix}$
(a) $abc\left( b - c \right)\left( c - a \right)\left( a - b \right)$
(b) $\left( b - c \right)\left( c - a \right)\left( a - b \right)$

(c) $\left( a + b + c \right)\left( b - c \right)\left( c - a \right)\left( a - b \right)$

(d) none of these

Q 27 | Page 95

If $x, y \in \mathbb{R}$, then the determinant

$∆ = \begin{vmatrix}\cos x & - \sin x & 1 \\ \sin x & \cos x & 1 \\ \cos\left( x + y \right) & - \sin\left( x + y \right) & 0\end{vmatrix}$
(a) $\left[ - \sqrt{2}, \sqrt{2} \right]$
(b) $\left[ - 1, 1 \right]$
(c)  $\left[ - \sqrt{2}, 1 \right]$
(d) $\left[ - 1, - \sqrt{2} \right]$
Q 28 | Page 96

The maximum value of  $∆ = \begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin\theta & 1 \\ 1 + \cos\theta & 1 & 1\end{vmatrix}$ is (θ is real)
(a) 1/2
(b) sqrt3/2
(c) sqrt2
(d) -sqrt3/2

Q 29 | Page 96

The value of the determinant $\begin{vmatrix}x & x + y & x + 2y \\ x + 2y & x & x + y \\ x + y & x + 2y & x\end{vmatrix}$ is
(a) 9x2(x + y)
(b) 9y2(x + y)
(c) 3y2(x + y)
(d) 7x2(x + y)

Q 30 | Page 96

Let $f\left( x \right) = \begin{vmatrix}\cos x & x & 1 \\ 2\sin x & x & 2x \\ \sin x & x & x\end{vmatrix}$ $\lim_{x \to 0} \frac{f\left( x \right)}{x^2}$  is equal to
(a) 0
(b) −1
(c) 2
(d) 3

Q 31 | Page 96

There are two values of a which makes the determinant  $∆ = \begin{vmatrix}1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2a\end{vmatrix}$  equal to 86. The sum of these two values is

(a) 4
(b) 5
(c) −4
(d) 9

Q 32 | Page 96

If $\begin{vmatrix}a & p & x \\ b & q & y \\ c & r & z\end{vmatrix} = 16$ , then the value of $\begin{vmatrix}p + x & a + x & a + p \\ q + y & b + y & b + q \\ r + z & c + z & c + r\end{vmatrix}$ is
(a) 4
(b) 8
(c) 16
(d) 32

Q 33 | Page 96

The value of $\begin{vmatrix}1 & 1 & 1 \\ {}^n C_1 & {}^{n + 2} C_1 & {}^{n + 4} C_1 \\ {}^n C_2 & {}^{n + 2} C_2 & {}^{n + 4} C_2\end{vmatrix}$ is
(a) 2
(b) 4
(c) 8
(d) n2

#### Pages 22 - 25

Q 1.1 | Page 22

Find the adjoint of the following matrix:
$\begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Q 1.2 | Page 22

Find the adjoint of the following matrix:
$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Q 1.3 | Page 22

Find the adjoint of the following matrix:
$\begin{bmatrix}\cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Q 1.4 | Page 22

Find the adjoint of the following matrix:

$\begin{bmatrix}1 & \tan \alpha/2 \\ - \tan \alpha/2 & 1\end{bmatrix}$
Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Q 2.1 | Page 22

Compute the adjoint of the following matrix:
$\begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

Q 2.2 | Page 22

Compute the adjoint of the following matrix:

$\begin{bmatrix}1 & 2 & 5 \\ 2 & 3 & 1 \\ - 1 & 1 & 1\end{bmatrix}$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

Q 2.3 | Page 22

Compute the adjoint of the following matrix:

$\begin{bmatrix}2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1\end{bmatrix}$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

Q 2.4 | Page 22

Compute the adjoint of the following matrix:

$\begin{bmatrix}2 & 0 & - 1 \\ 5 & 1 & 0 \\ 1 & 1 & 3\end{bmatrix}$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

Q 3 | Page 22

For the matrix

$A = \begin{bmatrix}1 & - 1 & 1 \\ 2 & 3 & 0 \\ 18 & 2 & 10\end{bmatrix}$ , show that A (adj A) = O.
Q 4 | Page 22

If  $A = \begin{bmatrix}- 4 & - 3 & - 3 \\ 1 & 0 & 1 \\ 4 & 4 & 3\end{bmatrix}$, show that adj A = A.

Q 5 | Page 23

If $A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}$ , show that adj A = 3AT.

Q 6 | Page 23

Find A (adj A) for the matrix  $A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & 2 & - 1 \\ - 4 & 5 & 2\end{bmatrix} .$

Q 7.1 | Page 23

Find the inverse of the following matrix:

$\begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}$
Q 7.2 | Page 23

Find the inverse of the following matrix:

$\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$
Q 7.3 | Page 23

Find the inverse of the following matrix:

$\begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}$
Q 7.4 | Page 23

Find the inverse of the following matrix:

$\begin{bmatrix}2 & 5 \\ - 3 & 1\end{bmatrix}$
Q 8.1 | Page 23

Find the inverse of the following matrix.
$\begin{bmatrix}1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2\end{bmatrix}$

Q 8.2 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}1 & 2 & 5 \\ 1 & - 1 & - 1 \\ 2 & 3 & - 1\end{bmatrix}$
Q 8.3 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2\end{bmatrix}$
Q 8.4 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{bmatrix}$
Q 8.5 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}0 & 1 & - 1 \\ 4 & - 3 & 4 \\ 3 & - 3 & 4\end{bmatrix}$
Q 8.6 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}0 & 0 & - 1 \\ 3 & 4 & 5 \\ - 2 & - 4 & - 7\end{bmatrix}$
Q 8.7 | Page 23

Find the inverse of the following matrix.

$\begin{bmatrix}1 & 0 & 0 \\ 0 & \cos \alpha & \sin \alpha \\ 0 & \sin \alpha & - \cos \alpha\end{bmatrix}$
Q 9.1 | Page 23

Find the inverse of the following matrix and verify that $A^{- 1} A = I_3$

$\begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}$
Q 9.2 | Page 23

Find the inverse of the following matrix and verify that $A^{- 1} A = I_3$

$\begin{bmatrix}2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2\end{bmatrix}$
Q 10.1 | Page 23

For the following pair of matrix verity that $\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :$ $A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 6 \\ 3 & 2\end{bmatrix}$

Q 10.2 | Page 23

For the following pair of matrix verity that $\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :$ $A = \begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 5 \\ 3 & 4\end{bmatrix}$

Q 11 | Page 23

Let $A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B = \begin{bmatrix}6 & 7 \\ 8 & 9\end{bmatrix} .\text{ Find }\left( AB \right)^{- 1}$

Q 12 | Page 23

Given $A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}$, compute A−1 and show that $2 A^{- 1} = 9I - A .$

Q 13 | Page 23

If $A = \begin{bmatrix}4 & 5 \\ 2 & 1\end{bmatrix}$ , then show that $A - 3I = 2 \left( I + 3 A^{- 1} \right) .$

Q 14 | Page 23

Find the inverse of the matrix $A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}$ and show that $a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .$

Q 15 | Page 23

Given  $A = \begin{bmatrix}5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1\end{bmatrix}, B^{- 1} = \begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}$ . Compute (AB)−1.

Q 16.1 | Page 23

Let
$F \left( \alpha \right) = \begin{bmatrix}\cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{bmatrix}\text{ and }G\left( \beta \right) = \begin{bmatrix}\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ - \sin \beta & 0 & \cos \beta\end{bmatrix}$

Show that

$\left[ F \left( \alpha \right) \right]^{- 1} = F \left( - \alpha \right)$
Q 16.2 | Page 23

Let
$F \left( \alpha \right) = \begin{bmatrix}\cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{bmatrix}\text{ and }G\left( \beta \right) = \begin{bmatrix}\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ - \sin \beta & 0 & \cos \beta\end{bmatrix}$

Show that

$\left[ G \left( \beta \right) \right]^{- 1} = G \left( - \beta \right)$
Q 16.3 | Page 23

Let
$F \left( \alpha \right) = \begin{bmatrix}\cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{bmatrix}\text{ and }G\left( \beta \right) = \begin{bmatrix}\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ - \sin \beta & 0 & \cos \beta\end{bmatrix}$

Show that

$\left[ G \left( \beta \right) \right]^{- 1} = G \left( - \beta \right)$
Q 17 | Page 23

If $A = \begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix}$ , verify that $A^2 - 4 A + I = O,\text{ where }I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\text{ and }O = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$ . Hence, find A−1.

Q 18 | Page 24

Show that

$A = \begin{bmatrix}- 8 & 5 \\ 2 & 4\end{bmatrix}$ satisfies the equation $A^2 + 4A - 42I = O$. Hence, find A−1.
Q 19 | Page 24

If $A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}$, show that

$A^2 - 5A + 7I = O$.  Hence, find A−1.
Q 20 | Page 24

If  $A = \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix}$, find x and y such that

$A^2 = xA + yI = O$ . Hence, evaluate A−1.
Q 21 | Page 24

If $A = \begin{bmatrix}3 & - 2 \\ 4 & - 2\end{bmatrix}$, find the value of $\lambda$  so that $A^2 = \lambda A - 2I$. Hence, find A−1.

Q 22 | Page 24

Show that $A = \begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix}$ satisfies the equation $x^2 - 3x - 7 = 0$. Thus, find A−1.

Q 23 | Page 24

Show that $A = \begin{bmatrix}6 & 5 \\ 7 & 6\end{bmatrix}$ satisfies the equation $x^2 - 12x + 1 = O$. Thus, find A−1.

Q 24 | Page 24

For the matrix $A = \begin{bmatrix}1 & 1 & 1 \\ 1 & 2 & - 3 \\ 2 & - 1 & 3\end{bmatrix}$ . Show that

$A^{- 3} - 6 A^2 + 5A + 11 I_3 = O$. Hence, find A−1.
Q 25 | Page 24

Show that the matrix, $A = \begin{bmatrix}1 & 0 & - 2 \\ - 2 & - 1 & 2 \\ 3 & 4 & 1\end{bmatrix}$  satisfies the equation,  $A^3 - A^2 - 3A - I_3 = O$ . Hence, find A−1.

Q 26 | Page 24
If $A = \begin{bmatrix}2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2\end{bmatrix}$.
Verify that $A^3 - 6 A^2 + 9A - 4I = O$  and hence find A−1.
Q 27 | Page 24
If $A = \frac{1}{9}\begin{bmatrix}- 8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & - 8 & 4\end{bmatrix}$,
prove that  $A^{- 1} = A^3$
Q 28 | Page 24

If $A = \begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}$ , show that $A^{- 1} = A^3$

Q 29 | Page 24

If $A = \begin{bmatrix}- 1 & 2 & 0 \\ - 1 & 1 & 1 \\ 0 & 1 & 0\end{bmatrix}$ , show that  $A^2 = A^{- 1} .$

Q 30 | Page 24

Solve the matrix equation $\begin{bmatrix}5 & 4 \\ 1 & 1\end{bmatrix}X = \begin{bmatrix}1 & - 2 \\ 1 & 3\end{bmatrix}$, where X is a 2 × 2 matrix.

Q 31 | Page 24

Find the matrix X satisfying the matrix equation $X\begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix} = \begin{bmatrix}14 & 7 \\ 7 & 7\end{bmatrix}$

Q 32 | Page 24

Find the matrix X for which

$\begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix} X \begin{bmatrix}- 1 & 1 \\ - 2 & 1\end{bmatrix} = \begin{bmatrix}2 & - 1 \\ 0 & 4\end{bmatrix}$

Q 33 | Page 24

Find the matrix X satisfying the equation

$\begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix} X \begin{bmatrix}5 & 3 \\ 3 & 2\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} .$
Q 34 | Page 24

If $A = \begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}$ , find $A^{- 1}$ and prove that $A^2 - 4A - 5I = O$

Q 36 | Page 25
$\text{ If }A^{- 1} = \begin{bmatrix}3 & - 1 & 1 \\ - 15 & 6 & - 5 \\ 5 & - 2 & 2\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 2 & - 2 \\ - 1 & 3 & 0 \\ 0 & - 2 & 1\end{bmatrix},\text{ find }\left( AB \right)^{- 1} .$
Q 37 | Page 25

If $A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1\end{bmatrix},\text{ find }\left( A^T \right)^{- 1} .$

Q 38 | Page 25

Find the adjoint of the matrix $A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}$  and hence show that $A\left( adj A \right) = \left| A \right| I_3$.

Q 39 | Page 25
$\text{ If }A = \begin{bmatrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{bmatrix},\text{ find }A^{- 1}\text{ and show that }A^{- 1} = \frac{1}{2}\left( A^2 - 3I \right) .$

#### Page 34

Q 1 | Page 34

Find the inverse $\begin{bmatrix}7 & 1 \\ 4 & - 3\end{bmatrix}$

Q 2 | Page 34

Find the inverse $\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}$

Q 3 | Page 34

Find the inverse $\begin{bmatrix}1 & 6 \\ - 3 & 5\end{bmatrix}$

Q 4 | Page 34

Find the inverse $\begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}$

Q 5 | Page 34

Find the inverse $\begin{bmatrix}3 & 10 \\ 2 & 7\end{bmatrix}$

Q 6 | Page 34

Find the inverse $\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}$

Q 7 | Page 34

Find the inverse $\begin{bmatrix}2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{bmatrix}$

Q 8 | Page 34

Find the inverse $\begin{bmatrix}2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2\end{bmatrix}$

Q 9 | Page 34

Find the inverse $\begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}$

Q 10 | Page 34

Find the inverse $\begin{bmatrix}1 & 2 & 0 \\ 2 & 3 & - 1 \\ 1 & - 1 & 3\end{bmatrix}$

Q 11 | Page 34

Find the inverse $\begin{bmatrix}2 & - 1 & 3 \\ 1 & 2 & 4 \\ 3 & 1 & 1\end{bmatrix}$

Q 12 | Page 34

Find the inverse $\begin{bmatrix}1 & 1 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1\end{bmatrix}$

Q 13 | Page 34

Find the inverse $\begin{bmatrix}2 & - 1 & 4 \\ 4 & 0 & 7 \\ 3 & - 2 & 7\end{bmatrix}$

Q 14 | Page 34

Find the inverse $\begin{bmatrix}3 & 0 & - 1 \\ 2 & 3 & 0 \\ 0 & 4 & 1\end{bmatrix}$

Q 15 | Page 34

Find the inverse $\begin{bmatrix}1 & 3 & - 2 \\ - 3 & 0 & - 1 \\ 2 & 1 & 0\end{bmatrix}$

Q 16 | Page 34

Find the inverse of each of the following matrices by using elementary row transformations:

$\begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}$

#### Pages 35 - 36

Q 1 | Page 35

Write the adjoint of the matrix $A = \begin{bmatrix}- 3 & 4 \\ 7 & - 2\end{bmatrix} .$

Q 2 | Page 35

If A is a square matrix such that A (adj A)  5I, where I denotes the identity matrix of the same order. Then, find the value of |A|.

Q 3 | Page 35

If A is a square matrix of order 3 such that |A| = 5, write the value of |adj A|.

Q 4 | Page 35

If A is a square matrix of order 3 such that |adj A| = 64, find |A|.

Q 5 | Page 35

If A is a non-singular square matrix such that |A| = 10, find |A−1|.

Q 6 | Page 35

If A, B, C are three non-null square matrices of the same order, write the condition on A such that AB = AC⇒ B = C.

Q 7 | Page 35

If A is a non-singular square matrix such that $A^{- 1} = \begin{bmatrix}5 & 3 \\ - 2 & - 1\end{bmatrix}$ , then find $\left( A^T \right)^{- 1} .$

Q 8 | Page 35

If adj $A = \begin{bmatrix}2 & 3 \\ 4 & - 1\end{bmatrix}\text{ and adj }B = \begin{bmatrix}1 & - 2 \\ - 3 & 1\end{bmatrix}$

Q 9 | Page 35

If A is symmetric matrix, write whether AT is symmetric or skew-symmetric.

Q 10 | Page 35

If A is a square matrix of order 3 such that |A| = 2, then write the value of adj (adj A).

Q 11 | Page 35

If A is a square matrix of order 3 such that |A| = 3, then write the value of adj (adj A).

Q 12 | Page 35

If A is a square matrix of order 3 such that adj (2A) = k adj (A), then write the value of k.

Q 13 | Page 35

If A is a square matrix, then write the matrix adj (AT) − (adj A)T.

Q 14 | Page 35

Let A be a 3 × 3 square matrix, such that A (adj A) = 2 I, where I is the identity matrix. Write the value of |adj A|.

Q 15 | Page 35

If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.

Q 16 | Page 35

If $A = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\text{ and }A \left( adj A = \right)\begin{bmatrix}k & 0 \\ 0 & k\end{bmatrix}$, then find the value of k.

Q 17 | Page 35

If A is an invertible matrix such that |A−1| = 2, find the value of |A|.

Q 18 | Page 35

If A is a square matrix such that $A \left( adj A \right) = \begin{bmatrix}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{bmatrix}$ , then write the value of |adj A|.

Q 19 | Page 35

If $A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}$ be such that $A^{- 1} = k A,$  then find the value of k.

Q 20 | Page 35

Let A be a square matrix such that $A^2 - A + I = O$, then write $A^{- 1}$  interms of A.

Q 21 | Page 36

If Cij is the cofactor of the element aij of the matrix $A = \begin{bmatrix}2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7\end{bmatrix}$, then write the value of a32C32.

Q 22 | Page 36

Find the inverse of the matrix $\begin{bmatrix}3 & - 2 \\ - 7 & 5\end{bmatrix} .$

Q 23 | Page 36

Find the inverse of the matrix $\begin{bmatrix}co \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}$

Q 24 | Page 36

If $A = \begin{bmatrix}1 & - 3 \\ 2 & 0\end{bmatrix}$, write adj A.

Q 25 | Page 36

If $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}, B = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ , find adj (AB).

Q 26 | Page 36

If $A = \begin{bmatrix}3 & 1 \\ 2 & - 3\end{bmatrix}$, then find |adj A|.

Q 27 | Page 36

If $A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}$ , write  $A^{- 1}$ in terms of A.

Q 28 | Page 36

Write $A^{- 1}\text{ for }A = \begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}$

Q 29 | Page 36

Use elementary column operation C2 → C2 + 2C1 in the following matrix equation : $\begin{bmatrix} 2 & 1 \\ 2 & 0\end{bmatrix} = \begin{bmatrix}3 & 1 \\ 2 & 0\end{bmatrix}\begin{bmatrix}1 & 0 \\ - 1 & 1\end{bmatrix}$

Q 30 | Page 36

In the following matrix equation use elementary operation R2 → R2 + Rand the equation thus obtained:

$\begin{bmatrix}2 & 3 \\ 1 & 4\end{bmatrix} \begin{bmatrix}1 & 0 \\ 2 & - 1\end{bmatrix} = \begin{bmatrix}8 & - 3 \\ 9 & - 4\end{bmatrix}$

#### Pages 37 - 39

Q 1 | Page 37

If A is an invertible matrix, then which of the following is not true
(a)$\left( A^2 \right)^{- 1} = \left( A^{- 1} \right)^2$
(b) $\left| A^{- 1} \right| = \left| A \right|^{- 1}$
(c) $\left( A^T \right)^{- 1} = \left( A^{- 1} \right)^T$
(d) $\left| A \right| \neq 0$

Q 2 | Page 37

If A is an invertible matrix of order 3, then which of the following is not true
(a) $\left| adj A \right| = \left| A \right|^2$
(b) $\left( A^{- 1} \right)^{- 1} = A$
(c) If $BA = CA,\text{ than }B \neq C$ , where B and C are square matrices of order 3
(d) $\left( AB \right)^{- 1} = B^{- 1} A^{- 1} , where B \neq \left[ b_{ij} \right]_{3 \times 3} and \left| B \right| \neq 0$

Q 3 | Page 37

If $A = \begin{bmatrix}3 & 4 \\ 2 & 4\end{bmatrix}, B = \begin{bmatrix}- 2 & - 2 \\ 0 & - 1\end{bmatrix},\text{ then }\left( A + B \right)^{- 1} =$

(a) is a skew-symmetric matrix
(b) A−1 + B−1
(c) does not exist
(d) none of these

Q 4 | Page 37

If $S = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$, then adj A is
(a) $\begin{bmatrix}- d & - b \\ - c & a\end{bmatrix}$

(b) $\begin{bmatrix}d & - b \\ - c & a\end{bmatrix}$
(c) $\begin{bmatrix}d & b \\ c & a\end{bmatrix}$
(d) $\begin{bmatrix}d & c \\ b & a\end{bmatrix}$

Q 5 | Page 37

If A is a singular matrix, then adj A is
(a) non-singular
(b) singular
(c) symmetric
(d) not defined

Q 6 | Page 37

If A, B are two n × n non-singular matrices, then
(a) AB is non-singular
(b) AB is singular
(c) $\left( AB \right)^{- 1} A^{- 1} B^{- 1}$
(d) (AB)−1 does not exist

Q 7 | Page 37

If $A = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}$ , then the value of |adj A| is

(a) a27
(b) a9
(c) a6
(d) a2

Q 8 | Page 37

If $A = \begin{bmatrix}1 & 2 & - 1 \\ - 1 & 1 & 2 \\ 2 & - 1 & 1\end{bmatrix}$ , then ded (adj (adj A)) is
(a) 144
(b) 143
(c) 142
(d) 14

Q 9 | Page 37

If B is a non-singular matrix and A is a square matrix, then det (B−1 AB) is equal to
(a) Det (A−1)
(b) Det (B−1)
(c) Det (A)
(d) Det (B)

Q 10 | Page 37

For any 2 × 2 matrix, if $A \left( adj A \right) = \begin{bmatrix}10 & 0 \\ 0 & 10\end{bmatrix}$ , then |A| is equal to

(a) 20
(c) 100
(d) 10
(d) 0

Q 11 | Page 37

If A5 = O such that $A^n \neq I\text{ for }1 \leq n \leq 4,\text{ then }\left( I - A \right)^{- 1}$
(a) A4
(b) A3
(c) I + A
(d) none of these

Q 12 | Page 37

If A satisfies the equation $x^3 - 5 x^2 + 4x + \lambda = 0$

(a) $x^3 - 5 x^2 + 4x + \lambda = 0$
(b) $\lambda \neq 2$
(c) $\lambda \neq 2$
(d) $\lambda \neq 0$

Q 13 | Page 37

If for the matrix A, A3 = I, then A−1 =
(a) A2
(b) A3
(c) A
(d) none of these

Q 14 | Page 38

If A and B are square matrices such that B = − A−1 BA, then (A + B)2 =
(a) O
(b) A2 + B2
(c) A2 + 2AB + B2
(d) A + B

Q 15 | Page 38

If $A = \begin{bmatrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{bmatrix},\text{ then }A^5 =$

(a) 5A
(b) 10A
(c) 16A
(d) 32A

Q 16 | Page 38

For non-singular square matrix A, B and C of the same order  $\left( A B^{- 1} C \right) =$
(a) $A^{- 1} B C^{- 1}$
(b) $C^{- 1} B^{- 1} A^{- 1}$
(c) $CB A^{- 1}$
(d) $C^{- 1} B A^{- 1}$

Q 17 | Page 38

The matrix $\begin{bmatrix}5 & 10 & 3 \\ - 2 & - 4 & 6 \\ - 1 & - 2 & b\end{bmatrix}$
(a) − 3
(b) 3
(c) 0
(d) non-existent

Q 18 | Page 38

If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is
(a) dn
(b) dn−1
(c) dn+1
(d) d

Q 19 | Page 38

If A is a matrix of order 3 and |A| = 8, then |adj A| =
(a) 1
(b) 2
(c) 23
(d) 26

Q 20 | Page 38

If $A^2 - A + I = 0$, then the inverse of A is
(a) A2
(b) A + I
(c) I − A
(d) A − I

Q 21 | Page 38

If A and B are invertible matrices, which of the following statement is not correct.
(a) $adj A = \left| A \right| A^{- 1}$
(b) $\det \left( A^{- 1} \right) = \left( \det A \right)^{- 1}$
(c) $\left( A + B \right)^{- 1} = A^{- 1} + B^{- 1}$
(d) $\left( AB \right)^{- 1} = B^{- 1} A^{- 1}$

Q 22 | Page 38

If A is a square matrix such that A2 = I, then A1 is equal to
(a) A + I
(b) A
(c) 0
(d) 2A

Q 23 | Page 38

Let $A = \begin{bmatrix}1 & 2 \\ 3 & - 5\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$ and X be a matrix such that A = BX, then X is equal to
(a) $\frac{1}{2}\begin{bmatrix}2 & 4 \\ 3 & - 5\end{bmatrix}$
(b) $\frac{1}{2}\begin{bmatrix}- 2 & 4 \\ 3 & 5\end{bmatrix}$
(c) $\begin{bmatrix}2 & 4 \\ 3 & - 5\end{bmatrix}$
(d) none of these.

Q 24 | Page 38

If $A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}$  be such that $A^{- 1} = kA$, then k equals

(a) 19
(b) 1/19
(c) − 19
(d) − 1/19

Q 25 | Page 38
If $A = \frac{1}{3}\begin{bmatrix}1 & 1 & 2 \\ 2 & 1 & - 2 \\ x & 2 & y\end{bmatrix}$  is orthogonal, then x + y =

(a) 3
(b) 0
(c) − 3
(d) 1

Q 26 | Page 38

If $A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 0 & 1 \\ a & b & 2\end{bmatrix},\text{ then aI + bA + 2 }A^2$ equals

(a) A
(b) − A
(c) ab A
(d) none of these

Q 27 | Page 38

If $\begin{bmatrix}1 & - \tan \theta \\ \tan \theta & 1\end{bmatrix} \begin{bmatrix}1 & \tan \theta \\ - \tan \theta & 1\end{bmatrix} - 1 = \begin{bmatrix}a & - b \\ b & a\end{bmatrix}$, then
(a) $a = 1, b = 1$
(b) $a = \cos 2 \theta, b = \sin 2 \theta$
(c) $a = \sin 2 \theta, b = \cos 2 \theta$
(d) none of these

Q 28 | Page 39

If a matrix A is such that 3

$A^3 + 2 A^2 + 5 A + I = 0,\text{ then }A^{- 1}$
(a) $- \left( 3 A^2 + 2 A + 5 \right)$
(b) $3 A^2 + 2 A + 5$
(c) $3 A^2 - 2 A - 5$
(d) none of these
Q 29 | Page 39

If A is an invertible matrix, then det (A1) is equal to
(a) $\det \left( A \right)$
(b) $\frac{1}{det \left( A \right)}$
(c) 1
(d) none of these

Q 30 | Page 39
If $A = \begin{bmatrix}2 & - 1 \\ 3 & - 2\end{bmatrix},\text{ then } A^n =$

(a) $A = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, if n is an even natural number
(b) $A = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ , if n is an odd natural number
(c) $A = \begin{bmatrix}- 1 & 0 \\ 0 & 1\end{bmatrix}, if n \in N$
(d) none of these

Q 31 | Page 39
If x, y, z are non-zero real numbers, then the inverse of the matrix $A = \begin{bmatrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{bmatrix}$, is
(a) $\begin{bmatrix}x^{- 1} & 0 & 0 \\ 0 & y^{- 1} & 0 \\ 0 & 0 & z^{- 1}\end{bmatrix}$
(b) $xyz \begin{bmatrix}x^{- 1} & 0 & 0 \\ 0 & y^{- 1} & 0 \\ 0 & 0 & z^{- 1}\end{bmatrix}$
(c) $\frac{1}{xyz}\begin{bmatrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{bmatrix}$
(d) $\frac{1}{xyz} \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$

#### Pages 14 - 18

Q 1.1 | Page 14

Solve the following system of equations by matrix method:
5x + 7y + 2 = 0
4x + 6y + 3 = 0

Q 1.2 | Page 14

Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5

Q 1.3 | Page 14

Solve the following system of equations by matrix method:
3x + 4y − 5 = 0
x − y + 3 = 0

Q 1.4 | Page 14

Solve the following system of equations by matrix method:
3x + y = 19
3x − y = 23

Q 1.5 | Page 14

Solve the following system of equations by matrix method:
3x + 7y = 4
x + 2y = −1

Q 1.6 | Page 14

Solve the following system of equations by matrix method:
3x + y = 7
5x + 3y = 12

Q 2.01 | Page 14

Solve the following system of equations by matrix method:
x + y − z = 3
2x + 3y + z = 10
3x − y − 7z = 1

Q 2.02 | Page 14

Solve the following system of equations by matrix method:
x + y + z = 3
2x − y + z = − 1
2x + y − 3z = − 9

Q 2.03 | Page 14

Solve the following system of equations by matrix method:
6x − 12y + 25z = 4
4x + 15y − 20z = 3
2x + 18y + 15z = 10

Q 2.04 | Page 14

Solve the following system of equations by matrix method:
3x + 4y + 7z = 14
2x − y + 3z = 4
x + 2y − 3z = 0

Q 2.05 | Page 14

Solve the following system of equations by matrix method:
$\frac{2}{x} - \frac{3}{y} + \frac{3}{z} = 10$
$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 10$
$\frac{3}{x} - \frac{1}{y} + \frac{2}{z} = 13$

Q 2.06 | Page 14

Solve the following system of equations by matrix method:
5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25

Q 2.07 | Page 14

Solve the following system of equations by matrix method:
3x + 4y + 2z = 8
2y − 3z = 3
x − 2y + 6z = −2

Q 2.08 | Page 14

Solve the following system of equations by matrix method:
2x + y + z = 2
x + 3y − z = 5
3x + y − 2z = 6

Q 2.09 | Page 14

Solve the following system of equations by matrix method:
2x + 6y = 2
3x − z = −8
2x − y + z = −3

Q 2.1 | Page 14

Solve the following system of equations by matrix method:
x − y + z = 2
2x − y = 0
2y − z = 1

Q 2.11 | Page 14

Solve the following system of equations by matrix method:
8x + 4y + 3z = 18
2x + y +z = 5
x + 2y + z = 5

Q 2.12 | Page 14

Solve the following system of equations by matrix method:
x + y + z = 6
x + 2z = 7
3x + y + z = 12

Q 2.13 | Page 14

Solve the following system of equations by matrix method:

$\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4, \frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1, \frac{6}{x} + \frac{9}{y} - \frac{20}{z} = 2; x, y, z \neq 0$

Q 2.14 | Page 14

Solve the following system of equations by matrix method:
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12

Q 3.1 | Page 15

Show that the following systems of linear equations is consistent and also find their solutions:
6x + 4y = 2
9x + 6y = 3

Q 3.2 | Page 15

Show that the following systems of linear equations is consistent and also find their solutions:
2x + 3y = 5
6x + 9y = 15

Q 3.3 | Page 15

Show that the following systems of linear equations is consistent and also find their solutions:
5x + 3y + 7z = 4
3x + 26y + 2z = 9
7x + 2y + 10z = 5

Q 3.4 | Page 15

Show that the following systems of linear equations is consistent and also find their solutions:
x − y + z = 3
2x + y − z = 2
−x −2y + 2z = 1

Q 3.5 | Page 15

Show that the following systems of linear equations is consistent and also find their solutions:
x + y + z = 6
x + 2y + 3z = 14
x + 4y + 7z = 30

Q 3.6 | Page 15

Show that the following systems of linear equations is consistent and also find their solutions:
2x + 2y − 2z = 1
4x + 4y − z = 2
6x + 6y + 2z = 3

Q 4.1 | Page 15

Show that each one of the following systems of linear equation is inconsistent:
2x + 5y = 7
6x + 15y = 13

Q 4.2 | Page 15

Show that each one of the following systems of linear equation is inconsistent:
2x + 3y = 5
6x + 9y = 10

Q 4.3 | Page 15

Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5

Q 4.4 | Page 15

Show that each one of the following systems of linear equation is inconsistent:
4x − 5y − 2z = 2
5x − 4y + 2z = −2
2x + 2y + 8z = −1

Q 4.5 | Page 15

Show that each one of the following systems of linear equation is inconsistent:
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3

Q 4.6 | Page 15

Show that each one of the following systems of linear equation is inconsistent:
x + y − 2z = 5
x − 2y + z = −2
−2x + y + z = 4

Q 5 | Page 15
If $A = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{bmatrix}\text{ and }B = \begin{bmatrix}2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5\end{bmatrix}$ are two square matrices, find AB and hence solve the system of linear equations: x − y = 3, 2x + 3y + 4z = 17, y + 2z = 7
Q 6 | Page 15

If $A = \begin{bmatrix}2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2\end{bmatrix}$, find A−1 and hence solve the system of linear equations 2x − 3y + 5z = 11, 3x + 2y − 4z = −5, x + y + 2z = −3

Q 7 | Page 16

Find A−1, if $A = \begin{bmatrix}1 & 2 & 5 \\ 1 & - 1 & - 1 \\ 2 & 3 & - 1\end{bmatrix}$ . Hence solve the following system of linear equations:x + 2y + 5z = 10, x − y − z = −2, 2x + 3y − z = −11

Q 8.1 | Page 16
If $A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1\end{bmatrix}$ , find A−1. Using A−1, solve the system of linear equations  x − 2y = 10, 2x + y + 3z = 8, −2y + z = 7.
Q 8.2 | Page 16
If $A = \begin{bmatrix}3 & - 4 & 2 \\ 2 & 3 & 5 \\ 1 & 0 & 1\end{bmatrix}$ , find A−1 and hence solve the following system of equations:
Q 8.3 | Page 16
$A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}7 & 2 & - 6 \\ - 2 & 1 & - 3 \\ - 4 & 2 & 5\end{bmatrix}$, find AB. Hence, solve the system of equations: x − 2y = 10, 2x + y + 3z = 8 and −2y + z = 7
Q 8.4 | Page 16

If $A = \begin{bmatrix}1 & 2 & 0 \\ - 2 & - 1 & - 2 \\ 0 & - 1 & 1\end{bmatrix}$ , find A−1. Using A−1, solve the system of linear equations   x − 2y = 10, 2x − y − z = 8, −2y + z = 7

Q 8.5 | Page 16

Given $A = \begin{bmatrix}2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5\end{bmatrix}, B = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{bmatrix}$ , find BA and use this to solve the system of equations  y + 2z = 7, x − y = 3, 2x + 3y + 4z = 17

Q 8.6 | Page 16

If $A = \begin{bmatrix}2 & 3 & 1 \\ 1 & 2 & 2 \\ 3 & 1 & - 1\end{bmatrix}$ , find A–1 and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8.

Q 8.7 | Page 16

Use product $\begin{bmatrix}1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4\end{bmatrix}\begin{bmatrix}- 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2\end{bmatrix}$  to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3.

Q 9 | Page 16

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.

Q 10 | Page 16

An amount of Rs 10,000 is put into three investments at the rate of 10, 12 and 15% per annum. The combined income is Rs 1310 and the combined income of first and  second investment is Rs 190 short of the income from the third. Find the investment in each using matrix method.

Q 11 | Page 16

A company produces three products every day. Their production on a certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using matrix method.

Q 12 | Page 16

The prices of three commodities P, Q and R are Rs x, y and z per unit respectively. A purchases 4 units of R and sells 3 units of P and 5 units of Q. B purchases 3 units of Q and sells 2 units of P and 1 unit of R. Cpurchases 1 unit of P and sells 4 units of Q and 6 units of R. In the process A, B and C earn Rs 6000, Rs 5000 and Rs 13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.

Q 13 | Page 17

The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

Q 14 | Page 17

A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.

Q 15 | Page 17

Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. xy and z respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of xy and z. What values are described in this equations?

Q 16 | Page 17

Two factories decided to award their employees for three values of (a) adaptable tonew techniques, (b) careful and alert in difficult situations and (c) keeping clam in tense situations, at the rate of ₹ x, ₹ y and ₹ z per person respectively. The first factory decided to honour respectively 2, 4 and 3 employees with a total prize money of ₹ 29000. The second factory decided to honour respectively 5, 2 and 3 employees with the prize money of ₹ 30500. If the three prizes per person together cost ₹ 9500, then
i) represent the above situation by matrix equation and form linear equation using matrix multiplication.
ii) Solve these equation by matrix method.
iii) Which values are reflected in the questions?

Q 17 | Page 17

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹1,600. School B wants to spend ₹2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

Q 18 | Page 17

Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹x each, ₹y each and ₹z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹1,000. School Q wants to spend ₹1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

Q 19 | Page 17

Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹2,200. School Q wants to spend ₹3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹1,200, using matrices, find the award money for each value.
Apart from these three values, suggest one more value which should be considered for award.

Q 20 | Page 18

A total amount of ₹7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and $8\frac{1}{2}$ % respectively. The total annual interest from these three accounts is ₹550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices.

Q 21 | Page 18

A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen.

Q 1 | Page 20

2x − y + z = 0
3x + 2y − z = 0
x + 4y + 3z = 0

Q 2 | Page 20

2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 5z = 0

Q 3 | Page 20

3x − y + 2z = 0
4x + 3y + 3z = 0
5x + 7y + 4z = 0

Q 4 | Page 20

x + y − 6z = 0
x − y + 2z = 0
−3x + y + 2z = 0

Q 5 | Page 20

x + y + z = 0
x − y − 5z = 0
x + 2y + 4z = 0

Q 6 | Page 20

x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0

Q 7 | Page 21

3x + y − 2z = 0
x + y + z = 0
x − 2y + z = 0

Q 8 | Page 21

2x + 3y − z = 0
x − y − 2z = 0
3x + y + 3z = 0

#### Page 21

Q 1 | Page 21
If $\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}1 \\ - 1 \\ 0\end{bmatrix}$, find x, y and z.
Q 2 | Page 21

If $\begin{bmatrix}1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}$, find x, y and z.

Q 3 | Page 21

If $\begin{bmatrix}1 & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ - 1 \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}$ , find x, y and z.

Q 4 | Page 21

Solve the following for x and y: $\begin{bmatrix}3 & - 4 \\ 9 & 2\end{bmatrix}\binom{x}{y} = \binom{10}{ 2}$

Q 5 | Page 21
If $\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}2 \\ - 1 \\ 3\end{bmatrix}$, find x, y, z.
Q 6 | Page 21
If $A = \begin{bmatrix}2 & 4 \\ 4 & 3\end{bmatrix}, X = \binom{n}{1}, B = \binom{ 8}{11}$  and AX = B, then find n.

#### Pages 21 - 23

Q 1 | Page 21

The system of equation x + y + z = 2, 3x − y + 2z = 6 and 3x + y + z = −18 has
(a) a unique solution
(b) no solution
(c) an infinite number of solutions
(d) zero solution as the only solution

Q 2 | Page 21

The number of solutions of the system of equations
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
is
(a) 3
(b) 2
(c) 1
(d) 0

Q 3 | Page 22

Let $X = \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}, A = \begin{bmatrix}1 & - 1 & 2 \\ 2 & 0 & 1 \\ 3 & 2 & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}3 \\ 1 \\ 4\end{bmatrix}$ . If AX = B, then X is equal to
(a) $\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}$
(b) $\begin{bmatrix}- 1 \\ - 2 \\ - 3\end{bmatrix}$
(c) $\begin{bmatrix}- 1 \\ - 2 \\ - 3\end{bmatrix}$
(d) $\begin{bmatrix}- 1 \\ 2 \\ 3\end{bmatrix}$
(e) $\begin{bmatrix}0 \\ 2 \\ 1\end{bmatrix}$

Q 4 | Page 22

The number of solutions of the system of equations:
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
(a) 3
(b) 2
(c) 1
(d) 0

Q 5 | Page 22

The system of linear equations:
x + y + z = 2
2x + y − z = 3
3x + 2y + kz = 4 has a unique solution if
(a) k ≠ 0
(b) −1 < k < 1
(c) −2 < k < 2
(d) k = 0

Q 6 | Page 22

Consider the system of equations:
a1x + b1y + c1z = 0
a2x + b2y + c2z = 0
a3x + b3y + c3z = 0,
if $\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}$= 0, then the system has
(a) more than two solutions
(b) one trivial and one non-trivial solutions
(c) no solution
(d) only trivial solution (0, 0, 0)

Q 7 | Page 22

Let a, b, c be positive real numbers. The following system of equations in x, y and z

$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1, \frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, - \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 has$
(a) no solution
(b) unique solution
(c) infinitely many solutions
(d) finitely many solutions
Q 8 | Page 22

For the system of equations:
x + 2y + 3z = 1
2x + y + 3z = 2
5x + 5y + 9z = 4
(a) there is only one solution
(b) there exists infinitely many solution
(c) there is no solution
(d) none of these

Q 9 | Page 22

The existence of the unique solution of the system of equations:
x + y + z = λ
5x − y + µz = 10
2x + 3y − z = 6
depends on
(a) µ only
(b) λ only
(c) λ and µ both
(d) neither λ nor µ

Q 10 | Page 23

The system of equations:
x + y + z = 5
x + 2y + 3z = 9
x + 3y + λz = µ
has a unique solution, if
(a) λ = 5, µ = 13
(b) λ ≠ 5
(c) λ = 5, µ ≠ 13
(d) µ ≠ 13

## R.D. Sharma solutions for Class 12 Mathematics chapter 7 - Adjoint and Inverse of a Matrix

R.D. Sharma solutions for Class 12 Mathematics chapter 7 (Adjoint and Inverse of a Matrix) include all questions with solution and detail explanation from Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session). This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has created the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 7 Adjoint and Inverse of a Matrix are Minors and Co-factors, Area of a Triangle, Introduction of Determinant, Determinants of Matrix of Order One and Two, Determinant of a Square Matrix, Properties of Determinants, Adjoint and Inverse of a Matrix, Elementary Transformations, Applications of Determinants and Matrices, Determinant of a Matrix of Order 3 × 3, Rule A=KB.

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