Advertisements
Advertisements
प्रश्न
Find the inverse of the matrix \[A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}\] and show that \[a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .\]
Advertisements
उत्तर
We have,
\[A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}\]
\[\text{ So, }adj(A) = \begin{bmatrix}\frac{1 + bc}{a} & - b \\ - c & a\end{bmatrix}\]
\[\text{ and }\left| A \right| = 1\]
\[ \therefore A^{- 1} = \begin{bmatrix}\frac{1 + bc}{a} & - b \\ - c & a\end{bmatrix}\]
\[\text{ Now, }a A^{- 1} = \left( a^2 + bc + 1 \right)I - aA\]
\[\text{ LHS }= a A^{- 1} = a\begin{bmatrix}\frac{1 + bc}{a} & - b \\ - c & a\end{bmatrix} = \begin{bmatrix}1 + bc & - ba \\ - ca & a^2\end{bmatrix}\]
\[\text{ RHS }= \left( a^2 + bc + 1 \right)I - aA\]
\[ = a^2 + bc + 1\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} - \begin{bmatrix}a^2 & ba \\ ca & 1 + bc\end{bmatrix}\]
\[ = \begin{bmatrix}a^2 + bc + 1 & 0 \\ 0 & a^2 + bc + 1\end{bmatrix} - \begin{bmatrix}a^2 & ba \\ ca & 1 + bc\end{bmatrix}\]
\[ = \begin{bmatrix}1 + bc & - ba \\ - ca & a^2\end{bmatrix} =\text{ LHS }\]
Hence proved .
APPEARS IN
संबंधित प्रश्न
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,-1,2),(2,3,5),(-2,0,1)]`
Find the inverse of the matrices (if it exists).
`[(2,-2),(4,3)]`
Find the inverse of the matrices (if it exists).
`[(1,0,0),(0, cos alpha, sin alpha),(0, sin alpha, -cos alpha)]`
Find the adjoint of the following matrix:
\[\begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}\]
Find the adjoint of the following matrix:
\[\begin{bmatrix}a & b \\ c & d\end{bmatrix}\]
Find the adjoint of the following matrix:
\[\begin{bmatrix}\cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}\]
Compute 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 and verify that \[A^{- 1} A = I_3\]
Given \[A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}\], compute A−1 and show that \[2 A^{- 1} = 9I - A .\]
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.
For the matrix \[A = \begin{bmatrix}1 & 1 & 1 \\ 1 & 2 & - 3 \\ 2 & - 1 & 3\end{bmatrix}\] . Show that
Find the matrix X for which
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}7 & 1 \\ 4 & - 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 2 & 0 \\ 2 & 3 & - 1 \\ 1 & - 1 & 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & - 1 & 4 \\ 4 & 0 & 7 \\ 3 & - 2 & 7\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 3 & - 2 \\ - 3 & 0 & - 1 \\ 2 & 1 & 0\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
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 a square matrix, then write the matrix adj (AT) − (adj A)T.
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 \[\begin{bmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\]
If A is an invertible matrix, then which of the following is not true ?
If \[A = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\] , then the value of |adj A| is _____________ .
If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is _____________ .
If a matrix A is such that \[3A^3 + 2 A^2 + 5 A + I = 0,\text{ then }A^{- 1}\] equal to _______________ .
|adj. A| = |A|2, where A is a square matrix of order two.
Find the adjoint of the matrix A, where A `= [(1,2,3),(0,5,0),(2,4,3)]`
Find the value of x for which the matrix A `= [(3 - "x", 2, 2),(2,4 - "x", 1),(-2,- 4,-1 - "x")]` is singular.
For what value of x, matrix `[(6-"x", 4),(3-"x", 1)]` is a singular matrix?
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 ____________.
A and B are invertible matrices of the same order such that |(AB)-1| = 8, If |A| = 2, then |B| is ____________.
