Advertisements
Advertisements
Question
Find the inverse of the following matrix.
Advertisements
Solution
\[ E = \begin{bmatrix}0 & 1 & - 1 \\ 4 & - 3 & 4 \\ 3 & - 3 & 4\end{bmatrix}\]
Now,
\[ C_{11} = \begin{vmatrix}- 3 & 4 \\ - 3 & 4\end{vmatrix} = 0, C_{12} = - \begin{vmatrix}4 & 4 \\ 3 & 4\end{vmatrix} = - 4\text{ and }C_{13} = \begin{vmatrix}4 & - 3 \\ 3 & - 3\end{vmatrix} = - 3\]
\[ C_{21} = - \begin{vmatrix}1 & - 1 \\ - 3 & 4\end{vmatrix} = - 1, C_{22} = \begin{vmatrix}0 & - 1 \\ 3 & 4\end{vmatrix} = 3\text{ and }C_{23} = - \begin{vmatrix}0 & 1 \\ 3 & - 3\end{vmatrix} = 3\]
\[ C_{31} = \begin{vmatrix}1 & - 1 \\ - 3 & 4\end{vmatrix} = 1, C_{32} = - \begin{vmatrix}0 & - 1 \\ 4 & 4\end{vmatrix} = - 4\text{ and }C_{33} = \begin{vmatrix}0 & 1 \\ 4 & - 3\end{vmatrix} = - 4\]
\[adjE = \begin{bmatrix}0 & - 4 & - 3 \\ - 1 & 3 & 3 \\ 1 & - 4 & - 4\end{bmatrix}^T = \begin{bmatrix}0 & - 1 & 1 \\ - 4 & 3 & - 4 \\ - 3 & 3 & - 4\end{bmatrix}\]
\[and \left| E \right| = - 1\]
\[ \therefore E^{- 1} = - 1\begin{bmatrix}0 & - 1 & 1 \\ - 4 & 3 & - 4 \\ - 3 & 3 & - 4\end{bmatrix} = \begin{bmatrix}0 & 1 & - 1 \\ 4 & - 3 & 4 \\ 3 & - 3 & 4\end{bmatrix}\]
APPEARS IN
RELATED QUESTIONS
The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?
Find the adjoint of the matrices.
`[(1,2),(3,4)]`
Verify A(adj A) = (adj A)A = |A|I.
`[(2,3),(-4,-6)]`
Verify A(adj A) = (adj A)A = |A|I.
`[(1,-1,2),(3,0,-2),(1,0,3)]`
Find the inverse of the matrices (if it exists).
`[(-1,5),(-3,2)]`
Find the inverse of the matrices (if it exists).
`[(1,-1,2),(0,2,-3),(3,-2,4)]`
Find the inverse of the matrices (if it exists).
`[(1,0,0),(0, cos alpha, sin alpha),(0, sin alpha, -cos alpha)]`
Let A = `[(1, sin theta, 1),(-sin theta,1,sin theta),(-1, -sin theta, 1)]` where 0 ≤ θ ≤ 2π, then ______.
Find the adjoint of the following matrix:
\[\begin{bmatrix}\cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}\]
Find the adjoint of the following matrix:
Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Find the inverse of the following matrix.
Find the inverse of the following matrix.
Find the inverse of the following matrix.
For the following pair of matrix verify that \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :\]
\[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 6 \\ 3 & 2\end{bmatrix}\]
If \[A = \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix}\], find x and y such that
Verify that \[A^3 - 6 A^2 + 9A - 4I = O\] and 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:
\[\begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}\]
If adj \[A = \begin{bmatrix}2 & 3 \\ 4 & - 1\end{bmatrix}\text{ and adj }B = \begin{bmatrix}1 & - 2 \\ - 3 & 1\end{bmatrix}\]
If A is symmetric matrix, write whether AT is symmetric or skew-symmetric.
If A is an invertible matrix, 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 any 2 × 2 matrix, if \[A \left( adj A \right) = \begin{bmatrix}10 & 0 \\ 0 & 10\end{bmatrix}\] , then |A| is equal to ______ .
If A5 = O such that \[A^n \neq I\text{ for }1 \leq n \leq 4,\text{ then }\left( I - A \right)^{- 1}\] equals ________ .
If \[A^2 - A + I = 0\], then the inverse of A is __________ .
If A and B are invertible matrices, which of the following statement is not correct.
If A is an invertible matrix, then det (A−1) is equal to ____________ .
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)]`
(A3)–1 = (A–1)3, where A is a square matrix and |A| ≠ 0.
|A–1| ≠ |A|–1, where A is non-singular matrix.
A square matrix A is invertible if det A is equal to ____________.
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 = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]`, find A–1. Use A–1 to solve the following system of equations 2x − 3y + 5z = 11, 3x + 2y – 4z = –5, x + y – 2z = –3
Given that A is a square matrix of order 3 and |A| = –2, then |adj(2A)| is equal to ______.
