<s>[INST] Determinant of a $5\times 5$ matrix using properties Apparently the determinant of the following matrix can be transformed into the determinant of a $2\times 2$ matrix multiplied by a scalar. Would anyone show me how?
$$ \left( \begin{array}{ccccc} 0 & 0 & 1 & 0 & 0 \\
2 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 3 & 0 \\
0 & 0 & 0 & 0 & 4 \\
0 & 5 & 0 & 0 & 0\end{array} \right)$$
 [/INST] I agree with the rest that permutations of rows and columns is sufficient. 
Another way is by doing Laplace expansion.
\begin{align}  \begin{vmatrix} 0 & 0 & 1 & 0 & 0 \\
2 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 3 & 0 \\
0 & 0 & 0 & 0 & 4 \\
0 & 5 & 0 & 0 & 0\end{vmatrix} &= \begin{vmatrix} 
2 & 0  & 0 & 0 \\
0 & 0  & 3 & 0 \\
0 & 0  & 0 & 4 \\
0 & 5  & 0 & 0\end{vmatrix} \\ &=  2\begin{vmatrix} 
 0  & 3 & 0 \\
 0  & 0 & 4 \\
 5  & 0 & 0\end{vmatrix} \\ &= 
2(5)\begin{vmatrix} 
    3 & 0 \\
    0 & 4 \\
   \end{vmatrix} \end{align}
</s>