Advertisements
Advertisements
प्रश्न
Find the inverse of the matrices (if it exists).
`[(2,1,3),(4,-1,0),(-7,2,1)]`
Advertisements
उत्तर
Let A = `[(2,1,3),(4,-1,0),(-7,2,1)]`
Then |A| = `|(2,1,3),(4,-1,0),(-7,2,1)|`
= 2(−1 − 0) − 1(4 − 0) + 3(8 − 7)
= −2 − 4 + 3
= −3 ≠ 0
So, A is a non-singular matrix and therefore, it is invertible. Let Cij be cofactor of aij in A. Then the cofactors of elements of A are given by,
C11 = `(-1)^(1+1) |(-1,0), (2,1)|`
= 1 × (−1 − 0)
= −1
C12 = `(-1)^(1+2) |(4,0), (-7,1)|`
= (−1) × (4 + 0)
= −1 × 4
= −4
C13 = `(-1)^(1+3)|(4,-1),(-7,2)|`
= 1 × (8 − 7)
= 1 × 1
= 1
C21 = `(-1)^(2+1) |(1,3), (2,1)|`
= (−1) × (1 − 6)
= (−1) × (−5)
= 5
C22 = `(-1)^(2+2) |(2,3), (-7,1)|`
= 1 × (2 + 21)
= 1 × 23
= 23
C23 = `(-1)^(2+3) |(2,1), (-7,2)|`
= (−1) × (4 + 7)
= (−1) × 11
= −11
C31 = `(-1)^(3+1) |(1,3), (-1,0)|`
= 1 × (0 + 3)
= 1 × 3
= 3
C32 = `(-1)^(3+2) |(2,3), (4,0)|`
= (−1) × (0 − 12)
= (−1) × (−12)
= 12
C33 = `(-1)^(3+3)|(2,1), (4,-1)|`
= 1 × (−2 − 4)
= 1 × (−6)
= −6
∴ adj A = `[(-1,-4,1),(5,23,-11),(3,12,-6)] = [(-1,5,3),(-4,23,12),(1,-11,-6)]`
A−1 = `1/|A|` adj A
= `1/-3 [(-1,5,3),(-4,23,12),(1,-11,-6)]`
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 inverse of the matrices (if it exists).
`[(1,0,0),(3,3,0),(5,2,-1)]`
Find the inverse of the matrices (if it exists).
`[(1,-1,2),(0,2,-3),(3,-2,4)]`
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.
If A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A3 − 6A2 + 9A − 4I = 0 and hence find A−1.
If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is ______.
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:
\[\begin{bmatrix}a & b \\ c & d\end{bmatrix}\]
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 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 and verify that \[A^{- 1} A = I_3\]
For the following pair of matrix verify that \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :\]
\[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 6 \\ 3 & 2\end{bmatrix}\]
For the following pair of matrix verify that \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :\]
\[A = \begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 5 \\ 3 & 4\end{bmatrix}\]
Given \[A = \begin{bmatrix}5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1\end{bmatrix}, B^{- 1} = \begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}\] . Compute (AB)−1.
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.
Show that
If \[A = \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix}\], find x and y such that
Verify that \[A^3 - 6 A^2 + 9A - 4I = O\] and hence find A−1.
Find the matrix X for which
If \[A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1\end{bmatrix},\text{ find }\left( A^T \right)^{- 1} .\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & - 1 & 3 \\ 1 & 2 & 4 \\ 3 & 1 & 1\end{bmatrix}\]
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:
\[\begin{bmatrix}3 & 0 & - 1 \\ 2 & 3 & 0 \\ 0 & 4 & 1\end{bmatrix}\]
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 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.
Find the inverse of the matrix \[\begin{bmatrix}3 & - 2 \\ - 7 & 5\end{bmatrix} .\]
If \[A = \begin{bmatrix}3 & 1 \\ 2 & - 3\end{bmatrix}\], then find |adj A|.
If \[S = \begin{bmatrix}a & b \\ c & d\end{bmatrix}\], then adj A is ____________ .
If A, B are two n × n non-singular matrices, then __________ .
If A and B are invertible matrices, which of the following statement is not correct.
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 _____________ .
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.
