$\left(A|b\right)=\left(\begin{array}{c}{a}_{11}\phantom{\rule{1em}{0ex}}{a}_{12}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{1n}\\ {a}_{21}\phantom{\rule{1em}{0ex}}{a}_{22}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{2n}\\ \vdots \\ {a}_{m1}\phantom{\rule{1em}{0ex}}{a}_{m2}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{mn}\end{array}\right|\left.\begin{array}{c}{b}_{1}\\ {b}_{2}\\ \vdots \\ {b}_{n}\end{array}\right)$

If leading coefficients zero then should be columns or rows are swapped accordingly so that a divison by the leading coefficient is possible. The value of the determinant is correct if, after the transformations the lower triangular matrix is zero, and the elements of the main diagonal are all equal to 1.