Advertisements
Advertisements
प्रश्न
If \[A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1\end{bmatrix},\text{ find }\left( A^T \right)^{- 1} .\]
Advertisements
उत्तर
We know that (AT)−1 = (A−1)T.
\[A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1\end{bmatrix}\]
\[ A^{- 1} = \frac{1}{\left| A \right|}Adj . A\]
Now,
\[\left| A \right| = \begin{vmatrix}1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1\end{vmatrix}\]
\[ = 1\left( - 1 - 8 \right) - 2\left( - 8 + 3 \right)\]
\[ = - 9 + 10\]
\[ = 1\]
\[\text{ Now, to find Adj . A}\]
\[A_{11} = \left( - 1 \right)^{1 + 1} \left( - 9 \right) = - 9\]
\[ A_{12} = \left( - 1 \right)^{1 + 2} \left( 8 \right) = - 8\]
\[ A_{13} = \left( - 1 \right)^{1 + 3} \left( - 2 \right) = - 2\]
\[A_{21} = \left( - 1 \right)^{2 + 1} \left( - 8 \right) = 8\]
\[ A_{22} = \left( - 1 \right)^{2 + 2} \left( 7 \right) = 7 \]
\[ A_{23} = \left( - 1 \right)^{2 + 3} \left( - 2 \right) = 2\]
\[ A_{31} = \left( - 1 \right)^{3 + 1} \left( - 5 \right) = - 5\]
\[ A_{32} = \left( - 1 \right)^{3 + 2} \left( 4 \right) = - 4\]
\[ A_{33} = \left( - 1 \right)^{3 + 3} \left( - 1 \right) = - 1\]
Therefore,
\[Adj . A = \begin{bmatrix}- 9 & 8 & - 5 \\ - 8 & 7 & - 4 \\ - 2 & 2 & - 1\end{bmatrix}\]
Thus,
\[ A^{- 1} = \begin{bmatrix}- 9 & 8 & - 5 \\ - 8 & 7 & - 4 \\ - 2 & 2 & - 1\end{bmatrix} . \]
\[ \left( A^T \right)^{- 1} = \left( A^{- 1} \right)^T \]
\[ = \begin{bmatrix}- 9 & 8 & - 5 \\ - 8 & 7 & - 4 \\ - 2 & 2 & - 1\end{bmatrix}^T \]
\[ = \begin{bmatrix}- 9 & - 8 & - 2 \\ 8 & 7 & 2 \\ - 5 & - 4 & - 1\end{bmatrix}\]
\[\text{ Hence, }\left( A^T \right)^{- 1} = \begin{bmatrix}- 9 & - 8 & - 2 \\ 8 & 7 & 2 \\ - 5 & - 4 & - 1\end{bmatrix} .\]
APPEARS IN
संबंधित प्रश्न
Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. School A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively with a total award money of Rs 1,600. School B wants to spend Rs 2,300 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for an award.
If A is an invertible matrix of order 2, then det (A−1) is equal to ______.
Let A = `[(1, sin theta, 1),(-sin theta,1,sin theta),(-1, -sin theta, 1)]` where 0 ≤ θ ≤ 2π, then ______.
Find 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:
Find the inverse of the following matrix.
Find the inverse of the following matrix.
Find the inverse of the following matrix and verify that \[A^{- 1} A = I_3\]
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.
If \[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}\], show that
prove that \[A^{- 1} = A^3\]
If \[A = \begin{bmatrix}- 1 & 2 & 0 \\ - 1 & 1 & 1 \\ 0 & 1 & 0\end{bmatrix}\] , show that \[A^2 = A^{- 1} .\]
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 equation
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 6 \\ - 3 & 5\end{bmatrix}\]
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:
\[\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2\end{bmatrix}\]
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 a square matrix, then write the matrix adj (AT) − (adj A)T.
If \[A = \begin{bmatrix}1 & - 3 \\ 2 & 0\end{bmatrix}\], write adj A.
If \[A = \begin{bmatrix}3 & 1 \\ 2 & - 3\end{bmatrix}\], then find |adj A|.
If \[A = \begin{bmatrix}1 & 2 & - 1 \\ - 1 & 1 & 2 \\ 2 & - 1 & 1\end{bmatrix}\] , then ded (adj (adj A)) is __________ .
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 for the matrix A, A3 = I, then A−1 = _____________ .
If \[A^2 - A + I = 0\], then the inverse of A is __________ .
If A and B are invertible matrices, which of the following statement is not correct.
|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.
If for a square matrix A, A2 – A + I = 0, then A–1 equals ______.
