Advertisements
Advertisements
Question
Find the inverse of the following matrix and verify that \[A^{- 1} A = I_3\]
Advertisements
Solution
\[B = \begin{bmatrix}2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2\end{bmatrix}\]
Now,
\[ C_{11} = \begin{vmatrix}4 & 1 \\ 7 & 2\end{vmatrix} = 1, C_{12} = - \begin{vmatrix}3 & 1 \\ 3 & 2\end{vmatrix} = - 3\text{ and }C_{13} = \begin{vmatrix}3 & 4 \\ 3 & 7\end{vmatrix} = 9\]
\[ C_{21} = - \begin{vmatrix}3 & 1 \\ 7 & 2\end{vmatrix} = 1, C_{22} = \begin{vmatrix}2 & 1 \\ 3 & 2\end{vmatrix} = 1\text{ and } C_{23} = - \begin{vmatrix}2 & 3 \\ 3 & 7\end{vmatrix} = - 5\]
\[ C_{31} = \begin{vmatrix}3 & 1 \\ 4 & 1\end{vmatrix} = - 1, C_{32} = - \begin{vmatrix}2 & 1 \\ 3 & 1\end{vmatrix} = 1\text{ and }C_{33} = \begin{vmatrix}2 & 3 \\ 3 & 4\end{vmatrix} = - 1\]
\[adjB = \begin{bmatrix}1 & - 3 & 9 \\ 1 & 1 & - 5 \\ - 1 & 1 & - 1\end{bmatrix}^T = \begin{bmatrix}1 & 1 & - 1 \\ - 3 & 1 & 1 \\ 9 & - 5 & - 1\end{bmatrix}\]
\[\text{ and }\left| B \right| = 2\]
\[ B^{- 1} = \frac{1}{2}\begin{bmatrix}1 & 1 & - 1 \\ - 3 & 1 & 1 \\ 9 & - 5 & - 1\end{bmatrix}\]
\[\text{ Now, }B^{- 1} B = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} = I_3\]
APPEARS IN
RELATED QUESTIONS
Find the adjoint of the matrices.
`[(1,2),(3,4)]`
Find the inverse of the matrices (if it exists).
`[(2,-2),(4,3)]`
Find the inverse of the matrices (if it exists).
`[(2,1,3),(4,-1,0),(-7,2,1)]`
Find the inverse of the matrices (if it exists).
`[(1,-1,2),(0,2,-3),(3,-2,4)]`
If A = `[(3,1),(-1,2)]` show that A2 – 5A + 7I = 0. Hence, find A–1.
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 is an invertible matrix of order 2, then det (A−1) is equal to ______.
Compute the adjoint of the following matrix:
\[\begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}\]
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.
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 matrix \[A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}\] and show that \[a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .\]
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 equation
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}7 & 1 \\ 4 & - 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{bmatrix}\]
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 symmetric matrix, write whether AT is symmetric or skew-symmetric.
If A is an invertible matrix such that |A−1| = 2, find the value of |A|.
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] , write \[A^{- 1}\] in terms of A.
If A is an invertible matrix of order 3, then which of the following is not true ?
If A5 = O such that \[A^n \neq I\text{ for }1 \leq n \leq 4,\text{ then }\left( I - A \right)^{- 1}\] equals ________ .
If for the matrix A, A3 = I, then A−1 = _____________ .
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 _____________ .
(a) 3
(b) 0
(c) − 3
(d) 1
If \[\begin{bmatrix}1 & - \tan \theta \\ \tan \theta & 1\end{bmatrix} \begin{bmatrix}1 & \tan \theta \\ - \tan \theta & 1\end{bmatrix} - 1 = \begin{bmatrix}a & - b \\ b & a\end{bmatrix}\], then _______________ .
If a matrix A is such that \[3A^3 + 2 A^2 + 5 A + I = 0,\text{ then }A^{- 1}\] equal to _______________ .
Find A−1, if \[A = \begin{bmatrix}1 & 2 & 5 \\ 1 & - 1 & - 1 \\ 2 & 3 & - 1\end{bmatrix}\] . Hence solve the following system of linear equations:x + 2y + 5z = 10, x − y − z = −2, 2x + 3y − z = −11
An amount of Rs 10,000 is put into three investments at the rate of 10, 12 and 15% per annum. The combined income is Rs 1310 and the combined income of first and second investment is Rs 190 short of the income from the third. Find the investment in each using matrix method.
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 adjoint of the matrix A, where A `= [(1,2,3),(0,5,0),(2,4,3)]`
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 ____________.
If A = [aij] is a square matrix of order 2 such that aij = `{(1"," "when i" ≠ "j"),(0"," "when" "i" = "j"):},` then A2 is ______.
If A is a square matrix of order 3 and |A| = 5, then |adj A| = ______.
If for a square matrix A, A2 – A + I = 0, then A–1 equals ______.
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.
