Advertisements
Advertisements
प्रश्न
Find the adjoint of the following matrix:
\[\begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}\]
Advertisements
उत्तर १
Given below is the square matrix. Here, we will interchange the diagonal elements and change the signs of the off-diagonal elements.
\[\ A = \begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}\]
\[adjA = \begin{bmatrix}4 & - 5 \\ - 2 & - 3\end{bmatrix}\]
\[(adjA)A = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[\left| A \right| = - 22\]
\[\left| A \right|I = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[A(adjA) = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[ \therefore (adjA)A = \left| A \right|I = A(adjA)\]
Hence verified.
उत्तर २
Given below is the square matrix. Here, we will interchange the diagonal elements and change the signs of the off-diagonal elements.
\[\ A = \begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}\]
\[adjA = \begin{bmatrix}4 & - 5 \\ - 2 & - 3\end{bmatrix}\]
\[(adjA)A = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[\left| A \right| = - 22\]
\[\left| A \right|I = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[A(adjA) = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[ \therefore (adjA)A = \left| A \right|I = A(adjA)\]
Hence verified.
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.
`[(1,2),(3,4)]`
Verify A(adj A) = (adj A)A = |A|I.
`[(2,3),(-4,-6)]`
Find the inverse of the matrices (if it exists).
`[(1,-1,2),(0,2,-3),(3,-2,4)]`
Find the inverse of the matrices (if it exists).
`[(1,0,0),(0, cos alpha, sin alpha),(0, sin alpha, -cos alpha)]`
For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A2 + aA + bI = 0.
If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is ______.
Find A (adj A) for the matrix \[A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & 2 & - 1 \\ - 4 & 5 & 2\end{bmatrix} .\]
Find the inverse of the following matrix.
Find the inverse of the matrix \[A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}\] and show that \[a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .\]
If \[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}\], show that
For the matrix \[A = \begin{bmatrix}1 & 1 & 1 \\ 1 & 2 & - 3 \\ 2 & - 1 & 3\end{bmatrix}\] . Show that
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 matrix equation \[X\begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix} = \begin{bmatrix}14 & 7 \\ 7 & 7\end{bmatrix}\]
Find the matrix X satisfying the equation
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:
\[\begin{bmatrix}3 & 10 \\ 2 & 7\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 1 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
If A is a square matrix, then write the matrix adj (AT) − (adj A)T.
If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.
If A is an invertible matrix of order 3, then which of the following is not true ?
If A, B are two n × n non-singular matrices, then __________ .
If d is the determinant of a square matrix A of order n, then the determinant of its adjoint 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 \[A = \begin{bmatrix}2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2\end{bmatrix}\], find A−1 and hence solve the system of linear equations 2x − 3y + 5z = 11, 3x + 2y − 4z = −5, x + y + 2z = −3
|A–1| ≠ |A|–1, where A is non-singular matrix.
Find x, if `[(1,2,"x"),(1,1,1),(2,1,-1)]` is singular
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 ____________.
If A is a square matrix of order 3, |A′| = −3, then |AA′| = ______.
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.
