Answer
0
Work Step by Step
First, use cofactor expansion (Theorem 4.1) along the first row:
\[
\begin{vmatrix}
1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9
\end{vmatrix}
=
1 \cdot
\begin{vmatrix}
5 & 6\\8 & 9
\end{vmatrix}
-2 \cdot
\begin{vmatrix}
4 & 6\\7 & 9
\end{vmatrix}
+3 \cdot
\begin{vmatrix}
4 & 5\\7 & 8
\end{vmatrix}
\]
Then simplify the expansion:
$=1 \cdot (45-48) -2 \cdot (36-42) + 3 \cdot (32-35)$
$=-3+12-9$
$=0$
