If \(A^{-1} = \displaystyle\frac{1}{5}\begin{bmatrix}1 & 2\\ -2 & 1\end{bmatrix}\), then the matrix A is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\begin{bmatrix}1 & -2\\ 2 & 1\end{bmatrix}\)
The matrix A is the inverse of \(A^{-1}\), that is \(A = (A^{-1})^{-1}\).
Let \(M = \begin{bmatrix}1 & 2\\ -2 & 1\end{bmatrix}\), so that \(A^{-1} = \displaystyle\frac{1}{5}M\).
\(|M| = 1 \cdot 1 - 2 \cdot (-2) = 5\)
\(\displaystyle M^{-1} = \frac{1}{5}\begin{bmatrix}1 & -2\\ 2 & 1\end{bmatrix}\)
\(\displaystyle A = \left(\frac{1}{5}M\right)^{-1} = 5M^{-1}\)
\(\displaystyle A = \begin{bmatrix}1 & -2\\ 2 & 1\end{bmatrix}\)
Check: \(\displaystyle A\,A^{-1} = \frac{1}{5}\begin{bmatrix}5 & 0\\ 0 & 5\end{bmatrix} = I\).