Laplace Expansion

Definition

Let $C$ be a cofactor matrix of $\mathbf{A}$

Along the $j$th column gives

$\text{det}(\mathbf{A})={\sum }_{i=1}^n{a}_{ij}{C}_{ij}={\sum }_{i=1}^n{a}_{ij}(-1{)}^{i+j}{M}_{i,j}$

Along the $i$th row gives

$\text{det}(\mathbf{A})={\sum }_{j=1}^n{a}_{ij}{C}_{ij}={\sum }_{j=1}^n{a}_{ij}(-1{)}^{i+j}{M}_{i,j}$