Question:
If $A=\left[\begin{array}{rrr}1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9\end{array}\right]$, find $|\mathrm{A}|$.
Solution:
Let $A=\left[\begin{array}{rrr}1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9\end{array}\right]$
By expanding along the first row, we have:
$\begin{aligned}|A| &=1\left|\begin{array}{ll}1 & -3 \\ 4 & -9\end{array}\right|-1\left|\begin{array}{ll}2 & -3 \\ 5 & -9\end{array}\right|-2\left|\begin{array}{ll}2 & 1 \\ 5 & 4\end{array}\right| \\ &=1(-9+12)-1(-18+15)-2(8-5) \\ &=1(3)-1(-3)-2(3) \\ &=3+3-6 \\ &=6-6 \\ &=0 \end{aligned}$