Advertisements
Advertisements
प्रश्न
Find the adjoint of the matrices.
`[(1,-1,2),(2,3,5),(-2,0,1)]`
Advertisements
उत्तर
Let A = `[(1,-1,2),(2,3,5),(-2,0,1)]`
Let Cij be cofactors of aij in A.
C11 = `(-1)^(1 + 1) |(3,5),(0,1)|`
= 3 − 0
= 3
C12 = `(-1)^(1 + 2) |(2,5),(-2,1)|`
= −(2 + 10)
= −12
C13 = `(-1)^(1 + 3) |(2,3),(-2,0)|`
= 0 + 6
= 6
C21 = `(-1)^(2 + 1) |(-1,2),(0,1)|`
= −(−1 − 0)
= 1
C22 = `(-1)^(2 + 2) |(1,2),(-2,1)|`
= 1 + 4
= 5
C23 = `(-1)^(2 + 3) |(1,-1),(-2,0)|`
= −(0 − 2)
= 2
C31 = `(-1)^(3 + 1) |(-1,2),(3,5)|`
= −5 − 6
= −11
C32 = `(-1)^(3 + 2) |(1,2),(2,5)|`
= −(5 − 4)
= −l
C33 = `(-1)^(3 + 3) |(1,-1),(2,3)|`
= 3 + 2
= 5
∴ Adj A `= [(C_11,C_12,C_13),(C_21,C_22,C_23),(C_31,C_32,C_33)]^T`
= `[(3,-12,6),(1,5,2),(-11,-1,5)]^T`
= `[(3,1,-11),(-12,5,-1),(6,2,5)]`
APPEARS IN
संबंधित प्रश्न
Verify A(adj A) = (adj A)A = |A|I.
`[(2,3),(-4,-6)]`
Find the inverse of the matrices (if it exists).
`[(2,-2),(4,3)]`
Find the inverse of the matrices (if it exists).
`[(2,1,3),(4,-1,0),(-7,2,1)]`
Find the adjoint of the following matrix:
\[\begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}\]
For the matrix
Find A (adj A) for the matrix \[A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & 2 & - 1 \\ - 4 & 5 & 2\end{bmatrix} .\]
Find the inverse of the following matrix:
If \[A = \begin{bmatrix}4 & 5 \\ 2 & 1\end{bmatrix}\] , then show that \[A - 3I = 2 \left( I + 3 A^{- 1} \right) .\]
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
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.
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.
If \[A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1\end{bmatrix},\text{ find }\left( A^T \right)^{- 1} .\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 6 \\ - 3 & 5\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 3 & - 2 \\ - 3 & 0 & - 1 \\ 2 & 1 & 0\end{bmatrix}\]
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.
If A is an invertible matrix such that |A−1| = 2, find the value of |A|.
Find the inverse of the matrix \[\begin{bmatrix}3 & - 2 \\ - 7 & 5\end{bmatrix} .\]
If \[A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}, B = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] , find adj (AB).
If A is an invertible matrix, then which of the following is not true ?
If A is an invertible matrix of order 3, then which of the following is not true ?
If \[A = \begin{bmatrix}1 & 2 & - 1 \\ - 1 & 1 & 2 \\ 2 & - 1 & 1\end{bmatrix}\] , then ded (adj (adj A)) is __________ .
For non-singular square matrix A, B and C of the same order \[\left( A B^{- 1} C \right) =\] ______________ .
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
If A = `[(x, 5, 2),(2, y, 3),(1, 1, z)]`, xyz = 80, 3x + 2y + 10z = 20, ten A adj. A = `[(81, 0, 0),(0, 81, 0),(0, 0, 81)]`
|adj. A| = |A|2, where A is a square matrix of order two.
If A, B be two square matrices such that |AB| = O, then ____________.
A square matrix A is invertible if det A is equal to ____________.
Find the adjoint of the matrix A, where A `= [(1,2,3),(0,5,0),(2,4,3)]`
For matrix A = `[(2,5),(-11,7)]` (adj A)' is equal to:
If A is a square matrix of order 3, |A′| = −3, then |AA′| = ______.
If A is a square matrix of order 3 and |A| = 5, then |adj A| = ______.
If for a square matrix A, A2 – A + I = 0, then A–1 equals ______.
Given that A is a square matrix of order 3 and |A| = –2, then |adj(2A)| is equal to ______.
