You are given that
\(\begin{bmatrix} 2 & -1 \\ a & -1\end{bmatrix}\)
is the inverse matrix of
\(\begin{bmatrix} -1 & 1 \\ -3 & 2\end{bmatrix}\).
What is the value of \(a\)?
The answer is \(3\).
From what we are given, we know that
\[\begin{bmatrix} 2 & -1 \\ a & -1\end{bmatrix}
\begin{bmatrix} -1 & 1 \\ -3 & 2\end{bmatrix}
= \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}.\]
Now,
\[\begin{bmatrix} 2 & -1 \\ a & -1\end{bmatrix}
\begin{bmatrix} -1 & 1 \\ -3 & 2\end{bmatrix}=
\begin{bmatrix} 1 & 0 \\ -a+3 & a-2\end{bmatrix}.\]
Hence, we must have \(-a+3 = 0\) and \(a-2 = 1\), implying that \(a = 3\).