The determinant of a matrix can be computed as a cofactor expansion across any row.

For any fixed row $i$, $\det A=\sum_j a_{ij}C_{ij}$, where $C_{ij}=(-1)^{i+j}\det A_{\widehat i,\widehat j}$. The value is independent of the selected row. Choose one containing many zeros to reduce work. The expansion follows by grouping permutation terms by their selected column in row $i$.Copy