Advertisements
Advertisements
प्रश्न
If \[A = \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix}\], find x and y such that
Advertisements
उत्तर
\[A = \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix}\]
\[ \therefore A^2 = \begin{bmatrix}22 & 27 \\ 18 & 31\end{bmatrix}\]
Now,
\[ A^2 - xA + yI = O\]
\[ \Rightarrow \begin{bmatrix}22 & 27 \\ 18 & 31\end{bmatrix} - \begin{bmatrix}4x & 3x \\ 2x & 5x\end{bmatrix} + \begin{bmatrix}y & 0 \\ 0 & y\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}22 - 4x - y & 27 - 3x \\ 18 - 2x & 31 - 5x - y\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
Thus, we have
\[22 - 4x + y = 0, 27 - 3x = 0, 18 - 2x = 0\text{ and }31 - 5x + y = 0\]
\[ \Rightarrow - 3x = - 27\]
\[ \Rightarrow x = 9\]
\[\text{ On putting }x = 9\text{ in }22 - 4x + y = 0,\text{ we get }\]
\[22 - 36 + y = 0\]
\[ \Rightarrow - 14 = - y\]
\[ \Rightarrow y = 14\]
Now,
\[ A^2 - 9A + 14I = 0\]
\[ \Rightarrow A^2 - 9A = - 14I\]
\[ \Rightarrow A^{- 1} A^2 - 9A A^{- 1} = - 14I A^{- 1} \left[\text{ Pre - multiplying both sides by }A^{- 1} \right]\]
\[ \Rightarrow A - 9I = - 14 A^{- 1} \]
\[ \Rightarrow A^{- 1} = - \frac{1}{14}\left( A - 9I \right)\]
\[ \Rightarrow A^{- 1} = - \frac{1}{14}\left\{ \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix} - \begin{bmatrix}9 & 0 \\ 0 & 9\end{bmatrix} \right\} = \frac{1}{14}\begin{bmatrix}5 & - 3 \\ - 2 & 4\end{bmatrix}\]
APPEARS IN
संबंधित प्रश्न
Verify A(adj A) = (adj A)A = |A|I.
`[(2,3),(-4,-6)]`
Let A = `[(3,7),(2,5)]` and B = `[(6,8),(7,9)]`. Verify that (AB)−1 = B−1A−1.
For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A2 + aA + bI = 0.
If A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A3 − 6A2 + 9A − 4I = 0 and hence find A−1.
If A is an invertible matrix of order 2, then det (A−1) is equal to ______.
If A−1 = `[(3,-1,1),(-15,6,-5),(5,-2,2)]` and B = `[(1,2,-2),(-1,3,0),(0,-2,1)]`, find (AB)−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 ______.
Compute the adjoint of the following matrix:
Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
If \[A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}\] , 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 \[A^{- 1} A = I_3\]
Given \[A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}\], compute A−1 and show that \[2 A^{- 1} = 9I - A .\]
Let
\[F \left( \alpha \right) = \begin{bmatrix}\cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{bmatrix}\text{ and }G\left( \beta \right) = \begin{bmatrix}\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ - \sin \beta & 0 & \cos \beta\end{bmatrix}\]
Show that
Show that
Show that the matrix, \[A = \begin{bmatrix}1 & 0 & - 2 \\ - 2 & - 1 & 2 \\ 3 & 4 & 1\end{bmatrix}\] satisfies the equation, \[A^3 - A^2 - 3A - I_3 = O\] . Hence, find A−1.
prove that \[A^{- 1} = A^3\]
If \[A = \begin{bmatrix}- 1 & 2 & 0 \\ - 1 & 1 & 1 \\ 0 & 1 & 0\end{bmatrix}\] , show that \[A^2 = A^{- 1} .\]
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 inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}\]
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:
\[\begin{bmatrix}2 & - 1 & 3 \\ 1 & 2 & 4 \\ 3 & 1 & 1\end{bmatrix}\]
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 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 A5 = O such that \[A^n \neq I\text{ for }1 \leq n \leq 4,\text{ then }\left( I - A \right)^{- 1}\] equals ________ .
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 and B are invertible matrices, then which of the following is not correct?
(A3)–1 = (A–1)3, where A is a square matrix and |A| ≠ 0.
|adj. A| = |A|2, where A is a square matrix of order two.
If A is a square matrix of order 3, |A′| = −3, then |AA′| = ______.
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
If for a square matrix A, A2 – A + I = 0, then A–1 equals ______.
