Advertisements
Advertisements
प्रश्न
Find A (adj A) for the matrix \[A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & 2 & - 1 \\ - 4 & 5 & 2\end{bmatrix} .\]
Advertisements
उत्तर
\[A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & 2 & - 1 \\ - 4 & 5 & 2\end{bmatrix}\]
Now,
\[ C_{11} = \begin{vmatrix}2 & - 1 \\ 5 & 2\end{vmatrix} = 9, C_{12} = - \begin{vmatrix}0 & - 1 \\ - 4 & 2\end{vmatrix} = 4\text{ and }C_{13} = \begin{vmatrix}0 & 2 \\ - 4 & 5\end{vmatrix} = 8\]
\[ C_{21} = - \begin{vmatrix}- 2 & 3 \\ 5 & 2\end{vmatrix} = 19, C_{22} = \begin{vmatrix}1 & 3 \\ - 4 & 2\end{vmatrix} = 14\text{ and }C_{23} = - \begin{vmatrix}1 & - 2 \\ - 4 & 5\end{vmatrix} = 3\]
\[ C_{31} = \begin{vmatrix}- 2 & 3 \\ 2 & - 1\end{vmatrix} = - 4, C_{32} = - \begin{vmatrix}1 & 3 \\ 0 & - 1\end{vmatrix} = 1\text{ and }C_{33} = \begin{vmatrix}1 & - 2 \\ 0 & 2\end{vmatrix} = 2\]
\[adj A = \begin{bmatrix}9 & 4 & 8 \\ 19 & 14 & 3 \\ - 4 & 1 & 2\end{bmatrix}^T = \begin{bmatrix}9 & 19 & - 4 \\ 4 & 14 & 1 \\ 8 & 3 & 2\end{bmatrix}\]
\[ \therefore A(adj A) = \begin{bmatrix}25 & 0 & 0 \\ 0 & 25 & 0 \\ 0 & 0 & 25\end{bmatrix}\]
APPEARS IN
संबंधित प्रश्न
Find the adjoint of the matrices.
`[(1,2),(3,4)]`
For the matrix A = `[(1,1,1),(1,2,-3),(2,-1,3)]` show that A3 − 6A2 + 5A + 11 I = 0. Hence, find A−1.
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.
Find the inverse of the following matrix.
Find the inverse of the following matrix.
Find the inverse of the following matrix and verify that \[A^{- 1} A = I_3\]
Find the inverse of the matrix \[A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}\] and show that \[a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .\]
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 \[A = \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix}\], find x and y such that
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.
If \[A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1\end{bmatrix},\text{ find }\left( A^T \right)^{- 1} .\]
Find the adjoint of the matrix \[A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}\] and hence show that \[A\left( adj A \right) = \left| A \right| I_3\].
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}3 & 0 & - 1 \\ 2 & 3 & 0 \\ 0 & 4 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
If A is symmetric matrix, write whether AT is symmetric or skew-symmetric.
Find the inverse of the matrix \[\begin{bmatrix}3 & - 2 \\ - 7 & 5\end{bmatrix} .\]
If A is an invertible matrix of order 3, then which of the following is not true ?
If \[A = \begin{bmatrix}3 & 4 \\ 2 & 4\end{bmatrix}, B = \begin{bmatrix}- 2 & - 2 \\ 0 & - 1\end{bmatrix},\text{ then }\left( A + B \right)^{- 1} =\]
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 ______ .
The matrix \[\begin{bmatrix}5 & 10 & 3 \\ - 2 & - 4 & 6 \\ - 1 & - 2 & b\end{bmatrix}\] is a singular matrix, if the value of b is _____________ .
If \[A^2 - A + I = 0\], then the inverse of A is __________ .
If \[A = \begin{bmatrix}2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2\end{bmatrix}\], find A−1 and hence solve the system of linear equations 2x − 3y + 5z = 11, 3x + 2y − 4z = −5, x + y + 2z = −3
If A = `[(x, 5, 2),(2, y, 3),(1, 1, z)]`, xyz = 80, 3x + 2y + 10z = 20, ten A adj. A = `[(81, 0, 0),(0, 81, 0),(0, 0, 81)]`
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?
A and B are invertible matrices of the same order such that |(AB)-1| = 8, If |A| = 2, then |B| is ____________.
If A is a square matrix of order 3, |A′| = −3, then |AA′| = ______.
If for a square matrix A, A2 – A + I = 0, then A–1 equals ______.
A furniture factory uses three types of wood namely, teakwood, rosewood and satinwood for manufacturing three types of furniture, that are, table, chair and cot.
The wood requirements (in tonnes) for each type of furniture are given below:
| Table | Chair | Cot | |
| Teakwood | 2 | 3 | 4 |
| Rosewood | 1 | 1 | 2 |
| Satinwood | 3 | 2 | 1 |
It is found that 29 tonnes of teakwood, 13 tonnes of rosewood and 16 tonnes of satinwood are available to make all three types of furniture.
Using the above information, answer the following questions:
- Express the data given in the table above in the form of a set of simultaneous equations.
- Solve the set of simultaneous equations formed in subpart (i) by matrix method.
- Hence, find the number of table(s), chair(s) and cot(s) produced.
