Laplace expansion ($\det A={\sum }_j(-1{)}^{i+j}{A}_{ij}\det A^{(i,j)}$)

Expand the determinant along any fixed row $i$ using the displayed formula, where $A^{(i,j)}$ deletes row $i$ and column $j$. The signed matrix minors are its cofactors. Applying the same formula to the transpose gives the column expansion. The row formula follows from the Leibniz formula for determinants by grouping permutations according to the selected column in row $i$. For a lower triangular matrix, expanding along the first row reduces the determinant to $A_{11}$ times a smaller triangular determinant; induction gives the product of all diagonal entries, even when some vanish.