Answer
-18
Work Step by Step
First, use cofactor expansion (Theorem 4.1) along the first row:
\[
\begin{vmatrix}
1 & 2 & 3\\2 & 3 & 1\\3 & 1 & 2
\end{vmatrix}
=
1 \cdot
\begin{vmatrix}
3 & 1\\1 & 2
\end{vmatrix}
-2 \cdot
\begin{vmatrix}
2 & 1\\3 & 2
\end{vmatrix}
+3 \cdot
\begin{vmatrix}
2 & 3\\3 & 1
\end{vmatrix}
\]
Then simplify the expansion:
$=1 \cdot (6-1) -2 \cdot (4-3)+ 3 \cdot (2-9)$
$=5-2-21$
$=-18$
