Advertisements
Advertisements
प्रश्न
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] be such that \[A^{- 1} = k A,\] then find the value of k.
Advertisements
उत्तर
\[A = \begin{bmatrix} 2 & 3\\5 & - 2 \end{bmatrix}\]
\[ \therefore \left| A \right| = \begin{vmatrix} 2 & 3\\5 & - 2 \end{vmatrix} = - 14 - 15 = - 19 \]
\[\text{ The value is non - zero, so }A^{- 1} \text{ exists .} \]
\[\text{ By definition, we have }\]
\[ A^{- 1} A = I [\text{ I is the identity matrix}]\]
\[ \Rightarrow kA . A = I [\text{ Substituting }A^{- 1} = kA]\]
\[ \Rightarrow k\begin{bmatrix} 2 & 3\\5 & - 2 \end{bmatrix}\begin{bmatrix} 2 & 3\\5 & - 2 \end{bmatrix} = \begin{bmatrix} 1 & 0\\0 & 1 \end{bmatrix}\]
\[ \Rightarrow k\begin{bmatrix} 4 + 15 & 6 - 6\\10 - 10 & 15 + 4 \end{bmatrix} = \begin{bmatrix} 1 & 0\\0 & 1 \end{bmatrix}\]
\[ \Rightarrow k\begin{bmatrix} 19 & 0\\0 & 19 \end{bmatrix} = \begin{bmatrix} 1 & 0\\0 & 1 \end{bmatrix}\]
\[ \Rightarrow k = \frac{1}{19}\]
APPEARS IN
संबंधित प्रश्न
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).
`[(2,-2),(4,3)]`
Find the adjoint of the following matrix:
\[\begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}\]
Compute the adjoint of the following matrix:
Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
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 following matrix and verify that \[A^{- 1} A = I_3\]
For the following pair of matrix verify that \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :\]
\[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 6 \\ 3 & 2\end{bmatrix}\]
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.
Show that \[A = \begin{bmatrix}6 & 5 \\ 7 & 6\end{bmatrix}\] satisfies the equation \[x^2 - 12x + 1 = O\]. Thus, find A−1.
Verify that \[A^3 - 6 A^2 + 9A - 4I = O\] and hence find A−1.
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 matrix X satisfying the equation
If \[A = \begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}\] , find \[A^{- 1}\] and prove that \[A^2 - 4A - 5I = O\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}7 & 1 \\ 4 & - 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 6 \\ - 3 & 5\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}\]
If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.
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.
Find the inverse of the matrix \[\begin{bmatrix}3 & - 2 \\ - 7 & 5\end{bmatrix} .\]
If \[S = \begin{bmatrix}a & b \\ c & d\end{bmatrix}\], then adj A is ____________ .
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 = `[(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.
(A3)–1 = (A–1)3, where A is a square matrix and |A| ≠ 0.
`("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 `abs((2"x", -1),(4,2)) = abs ((3,0),(2,1))` then x is ____________.
