\(\begin{bmatrix} 1 & 0 \\ -2 & 1\end{bmatrix}^{-2}
= \begin{bmatrix} 1 & 0 \\ a & 1\end{bmatrix}.\)
What is \(a\)?
The answer is 4.
Note that by definition,
\(\begin{bmatrix} 1 & 0 \\ -2 & 1\end{bmatrix}^{-2} =
\left(\begin{bmatrix} 1 & 0 \\ -2 & 1\end{bmatrix}^2\right)^{-1} =
\begin{bmatrix} 1 & 0 \\ -4 & 1\end{bmatrix}^{-1} =
\begin{bmatrix} 1 & 0 \\ 4 & 1\end{bmatrix}.\)