Answer
-7
Work Step by Step
First, use cofactor expansion (Theorem 4.1) along the first row:
\[
\begin{vmatrix}
0 & 1 & -1\\2 & 3 & -2\\-1 & 3 & 0
\end{vmatrix}
= 0 -
1 \cdot
\begin{vmatrix}
2 & -2\\-1 & 0
\end{vmatrix}
-1 \cdot
\begin{vmatrix}
2 & 3\\-1 & 3
\end{vmatrix}
\]
Then simplify the expansion:
$=-1 \cdot (0-2) - 1 \cdot (6+3)$
$=2-9$
$=-7$
