cofactor expansion

Cofactor expansion is a way to directly calculate $\det A$ without row operations. The cofactor $C_{i,j}$ of $A$ is the determinant of a smaller matrix where the $i$-th row and $j$-th column of $A$ is removed. In addition, the cofactor needs to be negated if $i+j$ is odd. In other words, $C_{i,j}=(-1{)}^{i+j}\det (A(i|j))$.

To calculate $\det A$, choose a row or column with the most zeros and the greatest elements (this is to make calculation by hand easier), iterate over each row or column element $A_{i,j}$, calculate $A_{i,j}\cdot C_{i,j}$, then sum them together. If the cofactor matrix is not 2x2, repeat cofactor expansion recursively.