Advertisements
Advertisements
प्रश्न
Find the matrix X for which
Advertisements
उत्तर
\[Let A = \begin{bmatrix} 3 & 2\\7 & 5 \end{bmatrix}, B = \begin{bmatrix} - 1 & 1\\ - 2 & 1 \end{bmatrix} \text{ and }C = \begin{bmatrix} 2 & - 1\\0 & 4 \end{bmatrix}\]
Now,
\[\left| A \right| = \begin{vmatrix} 3 & 2\\7 & 5 \end{vmatrix} = 15 - 14 = 1 \]
\[\left| B \right| = \begin{vmatrix} - 1 & 1\\ - 2 & 1 \end{vmatrix} = - 1 + 2 = 1 \]
\[\text{ Since, }\left| A \right| \neq 0\text{ and }\left| B \right| \neq 0\]
\[\text{ Hence, A & B are invertible, so }A^{- 1}\text{ and }B^{- 1}\text{ exist }. \]
Cofactors of matrix A are
\[ A_{11} = 5 A_{12} = - 7 A_{21} = - 2 A_{22} = 3 \]
Now,
\[adj A = \begin{bmatrix} 5 & - 7\\ - 2 & 3 \end{bmatrix}T = \begin{bmatrix} 5 & - 2\\ - 7 & 3 \end{bmatrix}\]
\[ A^{- 1} = \frac{1}{\left| A \right|}adj A = \begin{bmatrix} 5 & - 2\\ - 7 & 3 \end{bmatrix} \]
Cofactors of matrix B are
\[ B_{11} = 1 B_{12} = 2 B_{21} = - 1 B_{22} = - 1\]
Now,
\[adj B = \begin{bmatrix} 1 & 2\\ - 1 & - 1 \end{bmatrix}^T = \begin{bmatrix} 1 & - 1\\ 2 & - 1 \end{bmatrix}\]
\[ B^{- 1} = \frac{1}{\left| B \right|}adj B = \begin{bmatrix} 1 & - 1\\ 2 & - 1 \end{bmatrix} \]
The given equation becomes AXB = C
\[ \Rightarrow \left( A^{- 1} A \right)X\left( B B^{- 1} \right) = A^{- 1} C B^{- 1} \]
\[ \Rightarrow \left( I \right)X\left( I \right) = A^{- 1} C B^{- 1} \]
\[ \Rightarrow X = \begin{bmatrix} 5 & - 2\\ - 7 & 3 \end{bmatrix}\begin{bmatrix} 2 & - 1\\0 & 4 \end{bmatrix}\begin{bmatrix} 1 & - 1\\ 2 & - 1 \end{bmatrix}\]
\[ \Rightarrow X = \begin{bmatrix} 5 & - 2\\ - 7 & 3 \end{bmatrix}\begin{bmatrix} 2 - 2 & - 2 + 1\\ 0 + 8 & 0 - 4 \end{bmatrix}\]
\[ \Rightarrow X = \begin{bmatrix} 5 & - 2\\ - 7 & 3 \end{bmatrix}\begin{bmatrix} 0 & - 1\\ 8 & - 4 \end{bmatrix}\]
\[ \Rightarrow X = \begin{bmatrix} - 16 & 3\\ 24 & - 5 \end{bmatrix}\]
APPEARS IN
संबंधित प्रश्न
Find the inverse of the matrices (if it exists).
`[(1,0,0),(0, cos alpha, sin alpha),(0, sin alpha, -cos alpha)]`
Let A = `[(3,7),(2,5)]` and B = `[(6,8),(7,9)]`. Verify that (AB)−1 = B−1A−1.
If A = `[(3,1),(-1,2)]` show that A2 – 5A + 7I = 0. Hence, find A–1.
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 the adjoint of the following matrix:
\[\begin{bmatrix}a & b \\ c & d\end{bmatrix}\]
Compute 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.
If \[A = \begin{bmatrix}- 4 & - 3 & - 3 \\ 1 & 0 & 1 \\ 4 & 4 & 3\end{bmatrix}\], show that adj A = A.
If \[A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}\] , show that adj A = 3AT.
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 following matrix:
Find the inverse of the following matrix.
Find the inverse of the following matrix and verify that \[A^{- 1} A = I_3\]
For the following pair of matrix verify that \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :\]
\[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 6 \\ 3 & 2\end{bmatrix}\]
Given \[A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}\], compute A−1 and show that \[2 A^{- 1} = 9I - A .\]
If \[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}\], show that
Verify that \[A^3 - 6 A^2 + 9A - 4I = O\] and hence find A−1.
prove that \[A^{- 1} = A^3\]
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:
\[\begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}\]
If A is a square matrix, then write the matrix adj (AT) − (adj A)T.
If \[S = \begin{bmatrix}a & b \\ c & d\end{bmatrix}\], then adj A is ____________ .
If A is a singular matrix, then adj A is ______.
If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is _____________ .
If \[A^2 - A + I = 0\], then the inverse of A is __________ .
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] be such that \[A^{- 1} = kA\], then k equals ___________ .
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)]`
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 `abs((2"x", -1),(4,2)) = abs ((3,0),(2,1))` then x 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 and |A| = 5, then |adj A| = ______.
If A = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]`, find A–1. Use A–1 to solve the following system of equations 2x − 3y + 5z = 11, 3x + 2y – 4z = –5, x + y – 2z = –3
| 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.
