Advertisements
Advertisements
प्रश्न
If \[A = \begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}\] , find \[A^{- 1}\] and prove that \[A^2 - 4A - 5I = O\]
Advertisements
उत्तर
\[A = \begin{bmatrix} 1 & 2 & 2\\2 & 1 & 2\\2 & 2 & 1 \end{bmatrix} \]
\[ \Rightarrow \left| A \right| = \begin{vmatrix} 1 & 2 & 2\\2 & 1 & 2\\2 & 2 & 1 \end{vmatrix} = 1\left( 1 - 4 \right) - 2\left( 2 - 4 \right) + 2\left( 4 - 2 \right) = - 3 + 4 + 4 = 5 \]
\[\text{ Since, }\left| A \right| \neq 0\]
Hence, A is invertible .
Now,
\[ A^2 = \begin{bmatrix} 1 & 2 & 2\\2 & 1 & 2\\2 & 2 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 & 2\\2 & 1 & 2\\2 & 2 & 1 \end{bmatrix} = \begin{bmatrix} 1 + 4 + 4 & 2 + 2 + 4 & 2 + 4 + 2\\2 + 2 + 4 & 4 + 1 + 4 & 4 + 2 + 2\\2 + 4 + 2 & 4 + 2 + 2 & 1 + 4 + 4 \end{bmatrix} = \begin{bmatrix} 9 & 8 & 8\\8 & 9 & 8\\8 & 8 & 9 \end{bmatrix}\]
\[\text{ Now, }A^2 - 4A - 5I = \begin{bmatrix} 9 & 8 & 8\\8 & 9 & 8\\8 & 8 & 9 \end{bmatrix} - 4\begin{bmatrix} 1 & 2 & 2\\2 & 1 & 2\\2 & 2 & 1 \end{bmatrix} - 5\begin{bmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 9 - 4 - 5 & 8 - 8 - 0 & 8 - 8 - 0\\8 - 8 - 0 & 9 - 4 - 5 & 8 - 8 - 0\\8 - 8 - 0 & 8 - 8 - 0 & 9 - 4 - 5 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0 \end{bmatrix} = O \]
\[ \Rightarrow A^2 - 4A - 5I = O [\text{ Proved }]\]
\[\text{ Again,} A^2 - 4A - 5I = O\]
\[ \Rightarrow A^{- 1} \left( A^2 - 4A - 5I \right) = A^{- 1} O [\text{ Pre - multiplying with }A^{- 1} ]\]
\[ \Rightarrow A^{- 1} A^2 - 4 A^{- 1} A - 5 A^{- 1} = O\]
\[ \Rightarrow A - 4I = 5 A^{- 1} \]
\[ \Rightarrow 5 A^{- 1} = \begin{bmatrix} 1 & 2 & 2\\2 & 1 & 2\\2 & 2 & 1 \end{bmatrix} - 4\begin{bmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 - 4 & 2 - 0 & 2 - 0\\2 - 0 & 1 - 4 & 2 - 0\\2 - 0 & 2 - 0 & 1 - 4 \end{bmatrix} = \begin{bmatrix} - 3 & 2 & 2\\ 2 & - 3 & 2\\ 2 & 2 & - 3 \end{bmatrix}\]
\[ \Rightarrow A^{- 1} = \frac{1}{5}\begin{bmatrix} - 3 & 2 & 2\\ 2 & - 3 & 2\\ 2 & 2 & - 3 \end{bmatrix}\]
APPEARS IN
संबंधित प्रश्न
Verify A(adj A) = (adj A)A = |A|I.
`[(1,-1,2),(3,0,-2),(1,0,3)]`
For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A2 + aA + bI = 0.
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.
Find the adjoint of the following matrix:
\[\begin{bmatrix}\cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}\]
Find 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.
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 the inverse of the following matrix.
Given \[A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}\], compute A−1 and show that \[2 A^{- 1} = 9I - A .\]
If \[A = \begin{bmatrix}4 & 5 \\ 2 & 1\end{bmatrix}\] , then show that \[A - 3I = 2 \left( I + 3 A^{- 1} \right) .\]
Show that \[A = \begin{bmatrix}6 & 5 \\ 7 & 6\end{bmatrix}\] satisfies the equation \[x^2 - 12x + 1 = O\]. Thus, find A−1.
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.
Verify that \[A^3 - 6 A^2 + 9A - 4I = O\] and hence find 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:
\[\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] be such that \[A^{- 1} = k A,\] then find the value of k.
If \[A = \begin{bmatrix}3 & 1 \\ 2 & - 3\end{bmatrix}\], then find |adj 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 \[S = \begin{bmatrix}a & b \\ c & d\end{bmatrix}\], then adj A is ____________ .
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 \[A^2 - A + I = 0\], then the inverse of A is __________ .
If \[A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 0 & 1 \\ a & b & 2\end{bmatrix},\text{ then aI + bA + 2 }A^2\] equals ____________ .
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 = `[(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)]`
|adj. A| = |A|2, where A is a square matrix of order two.
Find the adjoint of the matrix A, where A `= [(1,2,3),(0,5,0),(2,4,3)]`
If A is a square matrix of order 3, |A′| = −3, then |AA′| = ______.
If for a square matrix A, A2 – A + I = 0, then A–1 equals ______.
