Advertisements
Advertisements
प्रश्न
If \[A = \begin{bmatrix}4 & 5 \\ 2 & 1\end{bmatrix}\] , then show that \[A - 3I = 2 \left( I + 3 A^{- 1} \right) .\]
उत्तर
Now,
\[adj(A) = \begin{bmatrix}1 & - 5 \\ - 2 & 4\end{bmatrix}\]
\[\text{ and }\left| A \right| = - 6\]
\[ \therefore A^{- 1} = - \frac{1}{6}\begin{bmatrix}1 & - 5 \\ - 2 & 4\end{bmatrix}\]
\[\text{ Now, }A - 3I = I + 3 A^{- 1} \]
\[\text{ LHS }= A - 3I = \begin{bmatrix}4 & 5 \\ 2 & 1\end{bmatrix} - 3\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 5 \\ 2 & - 2\end{bmatrix}\]
\[\text{ RHS }= 2\left( I + 3 A^{- 1} \right) = 2\left\{ \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} - 3 \times \frac{1}{6}\begin{bmatrix}1 & - 5 \\ - 2 & 4\end{bmatrix} \right\} = 2\begin{bmatrix}0 . 5 & 2 . 5 \\ 1 & - 1\end{bmatrix} = \begin{bmatrix}1 & 5 \\ 2 & - 2\end{bmatrix} =\text{ LHS }\]
Hence proved .
APPEARS IN
संबंधित प्रश्न
Find the inverse of the matrices (if it exists).
`[(2,-2),(4,3)]`
Find the inverse of the matrices (if it exists).
`[(1,2,3),(0,2,4),(0,0,5)]`
Find the inverse of the matrices (if it exists).
`[(1,0,0),(3,3,0),(5,2,-1)]`
Find the adjoint of the following matrix:
Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
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.
Let \[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B = \begin{bmatrix}6 & 7 \\ 8 & 9\end{bmatrix} .\text{ Find }\left( AB \right)^{- 1}\]
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
Show that \[A = \begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix}\] satisfies the equation \[x^2 - 3x - 7 = 0\]. Thus, 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.
Show that the matrix, \[A = \begin{bmatrix}1 & 0 & - 2 \\ - 2 & - 1 & 2 \\ 3 & 4 & 1\end{bmatrix}\] satisfies the equation, \[A^3 - A^2 - 3A - I_3 = O\] . Hence, find A−1.
Verify that \[A^3 - 6 A^2 + 9A - 4I = O\] and hence find A−1.
Find the matrix X satisfying the equation
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 6 \\ - 3 & 5\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}3 & 10 \\ 2 & 7\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2\end{bmatrix}\]
Find the inverse of the matrix \[\begin{bmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\]
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] , write \[A^{- 1}\] in terms of A.
If A is an invertible matrix, then which of the following is not true ?
If A is an invertible matrix of order 3, then which of the following is not true ?
If B is a non-singular matrix and A is a square matrix, then det (B−1 AB) is equal to ___________ .
For non-singular square matrix A, B and C of the same order \[\left( A B^{- 1} C \right) =\] ______________ .
If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is _____________ .
If A and B are invertible matrices, which of the following statement is not correct.
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] be such that \[A^{- 1} = kA\], then k equals ___________ .
|adj. A| = |A|2, where A is a square matrix of order two.
Find the adjoint of the matrix A `= [(1,2),(3,4)].`
Find x, if `[(1,2,"x"),(1,1,1),(2,1,-1)]` is singular
For what value of x, matrix `[(6-"x", 4),(3-"x", 1)]` is a singular matrix?
If `abs((2"x", -1),(4,2)) = abs ((3,0),(2,1))` then x is ____________.
If A = `[(0, 1),(0, 0)]`, then A2023 is equal to ______.
If for a square matrix A, A2 – A + I = 0, then A–1 equals ______.
| To raise money for an orphanage, students of three schools A, B and C organised an exhibition in their residential colony, where they sold paper bags, scrap books and pastel sheets made by using recycled paper. Student of school A sold 30 paper bags, 20 scrap books and 10 pastel sheets and raised ₹ 410. Student of school B sold 20 paper bags, 10 scrap books and 20 pastel sheets and raised ₹ 290. Student of school C sold 20 paper bags, 20 scrap books and 20 pastel sheets and raised ₹ 440. |
Answer the following question:
- Translate the problem into a system of equations.
- Solve the system of equation by using matrix method.
- Hence, find the cost of one paper bag, one scrap book and one pastel sheet.
