= Laplace expansion
{c}
{title2=$\det A=\sum_j(-1)^{i+j}A_{ij}\det A^{(i,j)}$}
= Cofactor expansion
{synonym}
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.
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