Advertisements
Advertisements
Question
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
Options
4
8
16
32
Advertisements
Solution
\[\begin{vmatrix}p + x & a + x & a + p \\ q + y & b + y & b + q \\ r + z & c + z & c + r\end{vmatrix} = \begin{vmatrix}p & a & a \\ q & b & b \\ r & c & c\end{vmatrix} + \begin{vmatrix}p & a & p \\ q & b & q \\ r & c & r\end{vmatrix} + \begin{vmatrix}p & x & a \\ q & y & b \\ r & z & c\end{vmatrix} + \begin{vmatrix}p & x & p \\ q & y & q \\ r & z & r\end{vmatrix} + \begin{vmatrix}x & a & a \\ y & b & b \\ z & c & c\end{vmatrix} + \begin{vmatrix}x & a & p \\ y & b & q \\ z & c & r\end{vmatrix} + \begin{vmatrix}x & x & a \\ y & y & b \\ z & z & c\end{vmatrix} + \begin{vmatrix}x & x & p \\ y & y & q \\ z & z & r\end{vmatrix}\]
\[ = 0 + 0 + \begin{vmatrix}p & x & a \\ q & y & b \\ r & z & c\end{vmatrix} + 0 + 0 + \begin{vmatrix}x & a & p \\ y & b & q \\ z & c & r\end{vmatrix} + 0 + 0\]
\[ = \begin{vmatrix}p & x & a \\ q & y & b \\ r & z & c\end{vmatrix} + \begin{vmatrix}x & a & p \\ y & b & q \\ z & c & r\end{vmatrix}\]
\[ = 2\begin{vmatrix}a & p & x \\ b & q & y \\ c & r & z\end{vmatrix}\]
\[ = 2 \times 16 = 32\]
APPEARS IN
RELATED QUESTIONS
Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to ______.
Show that
\[\begin{vmatrix}\sin 10^\circ & - \cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ\end{vmatrix} = 1\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}0 & x & y \\ - x & 0 & z \\ - y & - z & 0\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{vmatrix}\]
\[If ∆ = \begin{vmatrix}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{vmatrix}, ∆_1 = \begin{vmatrix}1 & 1 & 1 \\ yz & zx & xy \\ x & y & z\end{vmatrix},\text{ then prove that }∆ + ∆_1 = 0 .\]
Prove that
\[\begin{vmatrix}- bc & b^2 + bc & c^2 + bc \\ a^2 + ac & - ac & c^2 + ac \\ a^2 + ab & b^2 + ab & - ab\end{vmatrix} = \left( ab + bc + ca \right)^3\]
Find the area of the triangle with vertice at the point:
(−1, −8), (−2, −3) and (3, 2)
Prove that :
9x + 5y = 10
3y − 2x = 8
6x + y − 3z = 5
x + 3y − 2z = 5
2x + y + 4z = 8
x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
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|.
If |A| = 2, where A is 2 × 2 matrix, find |adj A|.
Evaluate: \[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
If ω is a non-real cube root of unity and n is not a multiple of 3, then \[∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}\]
Let \[A = \begin{bmatrix}1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1\end{bmatrix},\text{ where 0 }\leq \theta \leq 2\pi . \text{ Then,}\]
The value of the determinant
Solve the following system of equations by matrix method:
5x + 7y + 2 = 0
4x + 6y + 3 = 0
Solve the following system of equations by matrix method:
Show that each one of the following systems of linear equation is inconsistent:
2x + 3y = 5
6x + 9y = 10
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.
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.
2x + 3y − z = 0
x − y − 2z = 0
3x + y + 3z = 0
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
For the system of equations:
x + 2y + 3z = 1
2x + y + 3z = 2
5x + 5y + 9z = 4
Find the inverse of the following matrix, using elementary transformations:
`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`
The value of x, y, z for the following system of equations x + y + z = 6, x − y+ 2z = 5, 2x + y − z = 1 are ______
Solve the following equations by using inversion method.
x + y + z = −1, x − y + z = 2 and x + y − z = 3
Solve the following system of equations by using inversion method
x + y = 1, y + z = `5/3`, z + x = `4/3`
If `|(2x, 5),(8, x)| = |(6, 5),(8, 3)|`, then find x
If ` abs((1 + "a"^2 "x", (1 + "b"^2)"x", (1 + "c"^2)"x"),((1 + "a"^2) "x", 1 + "b"^2 "x", (1 + "c"^2) "x"), ((1 + "a"^2) "x", (1 + "b"^2) "x", 1 + "c"^2 "x"))`, then f(x) is apolynomial of degree ____________.
If the system of equations x + ky - z = 0, 3x - ky - z = 0 & x - 3y + z = 0 has non-zero solution, then k is equal to ____________.
Using the matrix method, solve the following system of linear equations:
`2/x + 3/y + 10/z` = 4, `4/x - 6/y + 5/z` = 1, `6/x + 9/y - 20/z` = 2.
