<s>[INST] we need to compute determinant of matrix. If$$
A=\begin{pmatrix}-2 & 9 & -1 \\ 8 & 0 & 7\end{pmatrix}, \quad B=\begin{pmatrix}1 & 5 \\ 6 & 3 \\ 2 & 4\end{pmatrix},
$$
find $|AB|$ without finding $AB$.
 [/INST] By the Cauchy-Binet formula, we have 
\begin{align*}
det(AB) &= \begin{vmatrix}
-2 & 9\\
8 & 0 
\end{vmatrix}\begin{vmatrix}
1 & 5\\
6 & 3 
\end{vmatrix} + \begin{vmatrix}
9 & -1\\
0 & 7 
\end{vmatrix}\begin{vmatrix}
6 & 3\\
2 & 4 
\end{vmatrix} + \begin{vmatrix}
-2 & -1\\
8 & 7 
\end{vmatrix}\begin{vmatrix}
1 & 2\\
5 & 4 
\end{vmatrix}\\
\end{align*}
I think you could continue from here.
</s>