Find adjoint of the matrix
\[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \]
\(\operatorname{adj}A = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}\)
Find adjoint of the matrix
\[ \begin{bmatrix} 1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1 \end{bmatrix} \]
\(\operatorname{adj}A = \begin{bmatrix} 3 & 1 & -11 \\ -12 & 5 & -1 \\ 6 & 2 & 5 \end{bmatrix}\)
Verify that \(A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|I\) for
\[ A = \begin{bmatrix} 2 & 3 \\ -4 & -6 \end{bmatrix} \]
Verify that \(A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|I\) for
\[ A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix} \]
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix} \]
\(A^{-1} = \frac{1}{14}\begin{bmatrix} 3 & 2 \\ -4 & 2 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix} \]
\(A^{-1} = \frac{1}{13}\begin{bmatrix} 2 & -5 \\ 3 & -1 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix} \]
\(A^{-1} = \frac{1}{10}\begin{bmatrix} 10 & -10 & 2 \\ 0 & 5 & -4 \\ 0 & 0 & 2 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix} \]
\(A^{-1} = -\frac{1}{3}\begin{bmatrix} -3 & 0 & 0 \\ 3 & -1 & 0 \\ -9 & -2 & 3 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1 \end{bmatrix} \]
\(A^{-1} = -\frac{1}{3}\begin{bmatrix} -1 & 5 & 3 \\ -4 & 23 & 12 \\ 1 & -11 & -6 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} \]
\(A^{-1} = \begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha \\ 0 & \sin\alpha & -\cos\alpha \end{bmatrix} \]
\(A^{-1} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha \\ 0 & \sin\alpha & -\cos\alpha \end{bmatrix}\)
Let \(A = \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix}\) and \(B = \begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix}\). Verify that \((AB)^{-1} = B^{-1}A^{-1}\).
If \(A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}\), show that \(A^2 - 5A + 7I = O\). Hence, find \(A^{-1}\).
\(A^{-1} = \frac{1}{7}\begin{bmatrix} 2 & -1 \\ 1 & 3 \end{bmatrix}\)
For the matrix \(A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}\), find the numbers \(a\) and \(b\) such that \(A^2 + aA + bI = O\).
\(a = -4,\; b = 1\)
For the matrix
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 2 & -1 & 3 \end{bmatrix} \]
show that \(A^3 - 6A^2 + 5A + 11I = O\). Hence, find \(A^{-1}\).
\(A^{-1} = \frac{1}{11}\begin{bmatrix} -3 & 4 & 5 \\ 9 & -1 & -4 \\ 5 & -3 & -1 \end{bmatrix}\)
If
\[ A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix} \]
verify that \(A^3 - 6A^2 + 9A - 4I = O\) and hence find \(A^{-1}\).
\(A^{-1} = \frac{1}{4}\begin{bmatrix} 3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix}\)
Let \(A\) be a nonsingular square matrix of order \(3 \times 3\). Then \(|\operatorname{adj}A|\) is equal to
If \(A\) is an invertible matrix of order 2, then \(\det(A^{-1})\) is equal to