Advertisements
Advertisements
Question
Let \[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B = \begin{bmatrix}6 & 7 \\ 8 & 9\end{bmatrix} .\text{ Find }\left( AB \right)^{- 1}\]
Advertisements
Solution
Given:
\[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\]
\[B = \begin{bmatrix}6 & 7 \\ 8 & 9\end{bmatrix}\]
\[AB = \begin{bmatrix}34 & 39 \\ 82 & 94\end{bmatrix}\]
Now,
\[\left| AB \right| = - 2\]
\[\text{ Since, }\left| AB \right| \neq 0\]
\[\text{ Hence, AB is invertible . Let }C_{ij} \text{ be the cofactor of }a_{in}\text{ in AB = }\left[ a_{ij} \right]\]
\[ C_{11} = 94 , C_{12} = - 82, C_{21} = - 39\text{ and }C_{22} = 34\]
\[adj(AB) = \begin{bmatrix}94 & - 82 \\ - 39 & 34\end{bmatrix}^T = \begin{bmatrix}94 & - 39 \\ - 82 & 34\end{bmatrix}\]
\[ \therefore \left( AB \right)^{- 1} = - \frac{1}{2}\begin{bmatrix}94 & - 39 \\ - 82 & 34\end{bmatrix} = \begin{bmatrix}- 47 & \frac{39}{2} \\ 41 & - 17\end{bmatrix}\]
APPEARS IN
RELATED QUESTIONS
Find the adjoint of the matrices.
`[(1,2),(3,4)]`
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 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 ______.
If \[A = \begin{bmatrix}- 4 & - 3 & - 3 \\ 1 & 0 & 1 \\ 4 & 4 & 3\end{bmatrix}\], show that adj A = A.
If \[A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}\] , show that adj A = 3AT.
Find the inverse of the following matrix:
Find the inverse of the following matrix:
Find the inverse of the following matrix.
Find the inverse of the matrix \[A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}\] and show that \[a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .\]
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.
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.
Find the matrix X satisfying the matrix equation \[X\begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix} = \begin{bmatrix}14 & 7 \\ 7 & 7\end{bmatrix}\]
Find the matrix X for which
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 3 & - 2 \\ - 3 & 0 & - 1 \\ 2 & 1 & 0\end{bmatrix}\]
Find the inverse of the matrix \[\begin{bmatrix}3 & - 2 \\ - 7 & 5\end{bmatrix} .\]
If \[A = \begin{bmatrix}3 & 1 \\ 2 & - 3\end{bmatrix}\], then find |adj A|.
If A is a singular matrix, then adj A is ______.
If B is a non-singular matrix and A is a square matrix, then det (B−1 AB) 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 satisfies the equation \[x^3 - 5 x^2 + 4x + \lambda = 0\] then A-1 exists if _____________ .
For non-singular square matrix A, B and C of the same order \[\left( A B^{- 1} C \right) =\] ______________ .
If A is an invertible matrix, then det (A−1) is equal to ____________ .
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)]`
If A = `[(0, 1, 3),(1, 2, x),(2, 3, 1)]`, A–1 = `[(1/2, -4, 5/2),(-1/2, 3, -3/2),(1/2, y, 1/2)]` then x = 1, y = –1.
`("aA")^-1 = 1/"a" "A"^-1`, where a is any real number and A is a square matrix.
|A–1| ≠ |A|–1, where A is non-singular matrix.
If A, B be two square matrices such that |AB| = O, then ____________.
For A = `[(3,1),(-1,2)]`, then 14A−1 is given by:
If A is a square matrix of order 3, |A′| = −3, then |AA′| = ______.
If for a square matrix A, A2 – A + I = 0, then A–1 equals ______.
