Advertisements
Advertisements
प्रश्न
Prove that
\[\begin{vmatrix}- bc & b^2 + bc & c^2 + bc \\ a^2 + ac & - ac & c^2 + ac \\ a^2 + ab & b^2 + ab & - ab\end{vmatrix} = \left( ab + bc + ca \right)^3\]
Advertisements
उत्तर
\[∆ = \begin{vmatrix}- bc & b^2 + bc & c^2 + bc \\ a^2 + ac & - ac & c^2 + ac \\ a^2 + ab & b^2 + ab & - ab\end{vmatrix}\]
\[ = \frac{1}{abc}\begin{vmatrix}- abc & a b^2 + abc & a c^2 + abc \\ a^2 b + abc & - abc & c^2 b + abc \\ a^2 c + abc & b^2 c + abc & - abc\end{vmatrix} \left[\text{ Applying }R_1 \text{ to }aR_1 , R_2 \text{ to }bR_2\text{ and }R_3 \text{ to }cR_3\text{ and then dividing by abc }\right]\]
\[ = \frac{abc}{abc}\begin{vmatrix}- bc & ab + ac & ac + ab \\ ab + bc & - ac & cb + ab \\ ac + bc & bc + ac & - ab\end{vmatrix} \left[\text{ Taking out a, b and c common from the three columns }\right]\]
\[\begin{vmatrix}ab + bc + ca & ab + bc + ca & ab + bc + ca \\ ab + bc & - ac & cb + ab \\ ac + bc & bc + ac & - ab\end{vmatrix} \left[\text{ Applying }R_1 \text{ to }R_1 + R_2 + R_3 \right]\]
\[ = (ab + bc + ca)\begin{vmatrix}1 & 1 & 1 \\ ab + bc & - ac & cb + ab \\ ac + bc & bc + ac & - ab\end{vmatrix}\]
\[ = (ab + bc + ca)\begin{vmatrix}0 & 0 & 1 \\ 0 & - (ab + bc + ac) & cb + ab \\ ac + bc + ab & bc + ac + ab & - ab\end{vmatrix} \left[\text{ Applying }C_1 \text{ to }C_1 - C_3 \text{ and }C_2 \text{ to }C_2 - C_3 \right]\]
\[ = (ab + bc + ca)\begin{vmatrix}0 & - (ab + bc + ac) \\ ac + bc + ab & bc + ac + ab\end{vmatrix}\]
\[ = (ab + bc + ca)(ab + bc + ac )^2 \]
\[ = (ab + bc + ca )^3\]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Solve the system of the following equations:
`2/x+3/y+10/z = 4`
`4/x-6/y + 5/z = 1`
`6/x + 9/y - 20/x = 2`
Evaluate
\[∆ = \begin{vmatrix}0 & \sin \alpha & - \cos \alpha \\ - \sin \alpha & 0 & \sin \beta \\ \cos \alpha & - \sin \beta & 0\end{vmatrix}\]
\[∆ = \begin{vmatrix}\cos \alpha \cos \beta & \cos \alpha \sin \beta & - \sin \alpha \\ - \sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{vmatrix}\]
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|.
Find the value of x, if
\[\begin{vmatrix}3x & 7 \\ 2 & 4\end{vmatrix} = 10\] , find the value of x.
Evaluate the following determinant:
\[\begin{vmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}, where A, B, C \text{ are the angles of }∆ ABC .\]
Evaluate :
\[\begin{vmatrix}a & b + c & a^2 \\ b & c + a & b^2 \\ c & a + b & c^2\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{vmatrix}\]
\[\begin{vmatrix}b^2 + c^2 & ab & ac \\ ba & c^2 + a^2 & bc \\ ca & cb & a^2 + b^2\end{vmatrix} = 4 a^2 b^2 c^2\]
Show that x = 2 is a root of the equation
Using determinants show that the following points are collinear:
(3, −2), (8, 8) and (5, 2)
2x − y = 1
7x − 2y = −7
Prove that :
Prove that
Solve each of the following system of homogeneous linear equations.
2x + 3y + 4z = 0
x + y + z = 0
2x − y + 3z = 0
If \[A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ - 1 & 0\end{bmatrix}\] , find |AB|.
If \[A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and B} = \begin{bmatrix}1 & - 4 \\ 3 & - 2\end{bmatrix},\text{ find }|AB|\]
Find the value of x from the following : \[\begin{vmatrix}x & 4 \\ 2 & 2x\end{vmatrix} = 0\]
Write the value of the determinant \[\begin{vmatrix}2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x\end{vmatrix}\]
Evaluate: \[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]
If x ∈ N and \[\begin{vmatrix}x + 3 & - 2 \\ - 3x & 2x\end{vmatrix}\] = 8, then find the value of x.
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
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
If A = `[(1, 2, 0), (-2, -1, -2), (0, -1, 1)]`, find A−1. Using A−1, solve the system of linear equations x − 2y = 10, 2x − y − z = 8, −2y + z = 7.
x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
The number of solutions of the system of equations
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
is
The number of solutions of the system of equations:
2x + y − z = 7
x − 3y + 2z = 1
x + 4y − 3z = 5
Write the value of `|(a-b, b- c, c-a),(b-c, c-a, a-b),(c-a, a-b, b-c)|`
On her birthday Seema decided to donate some money to children of an orphanage home. If there were 8 children less, everyone would have got ₹ 10 more. However, if there were 16 children more, everyone would have got ₹ 10 less. Using the matrix method, find the number of children and the amount distributed by Seema. What values are reflected by Seema’s decision?
System of equations x + y = 2, 2x + 2y = 3 has ______
Solve the following equations by using inversion method.
x + y + z = −1, x − y + z = 2 and x + y − z = 3
The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹ 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹ 90. Whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is ₹ 70. Find the cost of each item per dozen by using matrices
The number of values of k for which the linear equations 4x + ky + 2z = 0, kx + 4y + z = 0 and 2x + 2y + z = 0 possess a non-zero solution is
If `|(x + a, beta, y),(a, x + beta, y),(a, beta, x + y)|` = 0, then 'x' is equal to
The greatest value of c ε R for which the system of linear equations, x – cy – cz = 0, cx – y + cz = 0, cx + cy – z = 0 has a non-trivial solution, is ______.
