Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Find the inverse of the matrices (if it exists).
`[(2,-2),(4,3)]`
If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is ______.
Let A = `[(1, sin theta, 1),(-sin theta,1,sin theta),(-1, -sin theta, 1)]` where 0 ≤ θ ≤ 2π, then ______.
Compute the adjoint of the following matrix:
Verify that (adj A) A = |A| I = A (adj A) for the above 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:
Find the inverse of the following matrix.
Given \[A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}\], compute A−1 and show that \[2 A^{- 1} = 9I - A .\]
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
Verify that \[A^3 - 6 A^2 + 9A - 4I = O\] and hence find A−1.
If A is symmetric matrix, write whether AT is symmetric or skew-symmetric.
If A is a square matrix, then write the matrix adj (AT) − (adj A)T.
If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.
Find the inverse of the matrix \[\begin{bmatrix}3 & - 2 \\ - 7 & 5\end{bmatrix} .\]
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 ___________ .
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 A satisfies the equation \[x^3 - 5 x^2 + 4x + \lambda = 0\] then A-1 exists if _____________ .
If for the matrix A, A3 = I, then A−1 = _____________ .
For non-singular square matrix A, B and C of the same order \[\left( A B^{- 1} C \right) =\] ______________ .
If \[A^2 - A + I = 0\], then the inverse of A is __________ .
Let \[A = \begin{bmatrix}1 & 2 \\ 3 & - 5\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}\] and X be a matrix such that A = BX, then X is equal to _____________ .
If \[A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 0 & 1 \\ a & b & 2\end{bmatrix},\text{ then aI + bA + 2 }A^2\] equals ____________ .
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 = `[(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.
If A and B are invertible matrices, then which of the following is not correct?
(A3)–1 = (A–1)3, where A is a square matrix and |A| ≠ 0.
|A–1| ≠ |A|–1, where A is non-singular matrix.
|adj. A| = |A|2, where A is a square matrix of order two.
Find the adjoint of the matrix A `= [(1,2),(3,4)].`
If the equation a(y + z) = x, b(z + x) = y, c(x + y) = z have non-trivial solutions then the value of `1/(1+"a") + 1/(1+"b") + 1/(1+"c")` is ____________.
If `abs((2"x", -1),(4,2)) = abs ((3,0),(2,1))` then x is ____________.
If A is a square matrix of order 3 and |A| = 5, then |adj A| = ______.
If A = `[(0, 1),(0, 0)]`, then A2023 is equal to ______.
Given that A is a square matrix of order 3 and |A| = –2, then |adj(2A)| is equal to ______.
