Answer
474
Work Step by Step
We are given:
$A=\begin{bmatrix}
1 & 2 & -1\\
2& 1 & 4
\end{bmatrix} \rightarrow A^T=\begin{bmatrix}
1 & 2 \\
2& 1 \\
-1 & 4
\end{bmatrix} $
$B=\begin{bmatrix}
2 & 1\\
5& -2\\
4 & 7
\end{bmatrix} \rightarrow B^T=\begin{bmatrix}
2 & 5 & 4\\
1& -2 & 7
\end{bmatrix} $
then $B^TA^T=\begin{bmatrix}
8 & 25\\
-10& 28
\end{bmatrix}$
Hence here:
$\det(A^TB^T)=8.28-25.(-10)=474$
