$\textbf{Three elementary properties of the determinant function}$ 1. If $\mathbf{A}^{\ast}$ matrix is obtained from a square matrix by swapping two rows or two columns, then $det(\mathbf{A}^{\ast}) = -\det(\mathbf{A})$
2. If $\mathbf{A}^{\ast}$ matrix is obtained by $\mathbf{A}$ multiplying the i-th row, or j-th column scalar c, then $\det (\mathbf{A^{\ast}}) = c\det (\mathbf{A})$
3. If $\mathbf{A^{\ast}}$ matrix is obtained by replacing the k-row of $A_{k}$ by $A_{k} + cA_{i}$, or k-column by $A^{k} + cA^{i}$ with $k \neq j$, then $\det (\mathbf{A^{\ast}}) = \det (\mathbf{A})$