Advertisements
Advertisements
प्रश्न
Verify A(adj A) = (adj A)A = |A|I.
`[(1,-1,2),(3,0,-2),(1,0,3)]`
Advertisements
उत्तर
Let A = `[(1,-1,2),(3,0,-2),(1,0,3)]`
|A| = 1[0 − 0] + 1[9 + 2] + 2[0 − 0]
= 1 × 11
= 11
A11 = `(-1)^(1 + 1) |(0,-2),(0,3)|`
= (−1)2 [0 − 0]
= 0
A12 = `(-1)^(1 + 2) |(3,-2),(1,3)|`
= (−1)3 [9 + 2]
= −11
A13 = `(-1)^(1 + 3) |(3,0),(1,0)|`
= (−1)4 [0 − 0]
= 0
A21 = `(-1)^(2 + 1) |(-1,2),(0,3)|`
= (−1)3 [−3 − 0]
= −1 × (−3)
= 3
A22 = `(- 1)^(2 + 2) |(1, 2),(1,3)|`
= (−1)4 [3 − 2]
= 1 × 1
= 1
A23 = `(-1)^(2+ 3) |(1,-1),(1,0)|`
= (−1)5 [0 + 1]
= −1
A31 = `(1)^(3 + 1) |(-1,2),(0,-2)|`
= (−1)4 [2 − 0]
= 1 × 2
= 2
A32 = `(-1)^(3 + 2) |(1,2),(3,-2)|`
= (−1)5 [−2 − 6]
= −1 × (−8)
= 8
A33 = `(-1)^(3 + 3) |(1,-1),(3,0)|`
= (−1)6 [0 + 3]
= 1 × 3
= 3
adj A = `[(0,-11,0),(3,1,-1),(2,8,3)] = [(0,3,2),(-11,1,8),(0,-1,3)]`
L.H.S. = A(adj A) = `[(1,-1,2),(3,0,-2),(1,0,3)] [(0,3,2),(-11,1,8),(0,-1,3)]`
= `[(1 xx 0 + (- 1) xx (- 11) + 2 xx 0, 1 xx 3 + (- 1) xx 1 + 2 xx (- 1), 1 xx 2 + (- 1) xx 8 + 2 xx 3),(3 xx 0 + 0 xx (- 11) + (- 2) xx 0, 3 xx 3 + 0 xx 1 + (- 2) xx (- 1), 3 xx 2 + 0 xx 8 + (- 2) xx 3),(1 xx 0 + 0 xx (- 11) + 3 xx 0, 1 xx 3 + 0 xx 1 + 3 xx (-1), 1 xx 2 + 0 xx 8 + 3 + 3)]`
= `[(0+11+0,3 - 1 - 2, 2 - 8 + 6),(0+0+0, 9 + 0 + 2, 6 + 0 - 6),(0 + 0 + 0, 3 + 0 - 3, 2 + 0 + 9)]`
= `[(11,0,0),(0,11,0),(0,0,11)]`
= `11[(1,0,0),(0,1,0),(0,0,1)]`
= 11 · I
= |A| · I
R.H.S. = (adj A)A `= [(0,3,2),(-11,1,8),(0,-1,3)][(1,-1,2),(3,0,-2),(1,0,3)]`
= `[(0xx3 + 3 xx 3 + 2 xx 1,0xx(-1) + 3 xx 0 + 2 xx 0,0 xx 2 + 3 xx (- 2) + 2 xx 3),(-11xx1 + 1 xx 3 + 8 xx 1, -11 xx (- 1) + 1 xx 0 + 8 xx 0,-11 xx 2 + 1 xx(- 2) + 8 xx3),(0 xx 1 + (- 1) xx 3 + 3 xx 1, 0xx(- 1) + (- 1) xx 0 + 3 xx 0, 0xx2 + (- 1)xx (- 2) + 3 xx 3)]`
`= [(0 + 9 + 2, 0 + 0 + 0, 0 - 6 + 6),(- 11 + 3 + 8, 11 + 0 + 0, - 22 - 2 + 24),(0 - 3 + 3, 0 + 0 + 0, 0 + 2 + 9)]`
= `[(11,0,0),(0,11,0),(0,0,11)]`
= `11 [(1,0,0),(0,1,0),(0,0,1)]`
= 11 · I
= |A| · I
Hence, A(adj A) = (adj A)A = |A|I
APPEARS IN
संबंधित प्रश्न
Find the inverse of the matrices (if it exists).
`[(2,-2),(4,3)]`
Find the inverse of the matrices (if it exists).
`[(-1,5),(-3,2)]`
If A = `[(3,1),(-1,2)]` show that A2 – 5A + 7I = 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.
Let A = `[(1,2,1),(2,3,1),(1,1,5)]` verify that
- [adj A]–1 = adj(A–1)
- (A–1)–1 = A
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.
For the matrix
If \[A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}\] , show that adj A = 3AT.
Find the inverse of the following matrix:
Find the inverse of the following matrix.
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 \[A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}\], compute A−1 and show that \[2 A^{- 1} = 9I - A .\]
Find the inverse of the matrix \[A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}\] and show that \[a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .\]
Show that
If \[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}\], show that
Show that \[A = \begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix}\] satisfies the equation \[x^2 - 3x - 7 = 0\]. Thus, find A−1.
Find the matrix X satisfying the equation
If \[A = \begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}\] , find \[A^{- 1}\] and prove that \[A^2 - 4A - 5I = O\]
If A is symmetric matrix, write whether AT is symmetric or skew-symmetric.
If \[A = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\text{ and }A \left( adj A = \right)\begin{bmatrix}k & 0 \\ 0 & k\end{bmatrix}\], then find the value of k.
Find the inverse of the matrix \[\begin{bmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\]
If \[A = \begin{bmatrix}1 & - 3 \\ 2 & 0\end{bmatrix}\], write adj A.
If A is an invertible matrix of order 3, then which of the following is not true ?
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 matrix A is such that \[3A^3 + 2 A^2 + 5 A + I = 0,\text{ then }A^{- 1}\] equal to _______________ .
Using matrix method, solve the following system of equations:
x – 2y = 10, 2x + y + 3z = 8 and -2y + z = 7
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.
|adj. A| = |A|2, where A is a square matrix of order two.
Find the value of x for which the matrix A `= [(3 - "x", 2, 2),(2,4 - "x", 1),(-2,- 4,-1 - "x")]` is singular.
For A = `[(3,1),(-1,2)]`, then 14A−1 is given by:
Given that A is a square matrix of order 3 and |A| = –2, then |adj(2A)| is equal to ______.
| 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.
