The online calculator calculates the value of the determinant of a NxN matrix with the gaussian algorithm and shows all calculation steps for the matrix transformation to echelon form.

Note: 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.

With the Gauss method, the determinant is so transformed that the elements of the lower triangle matrix become zero. To do this, you use the row-factor rules and the addition of rows. The addition of rows does not change the value of the determinate. Factors of a row must be considered as multipliers before the determinat. If the determinat is triangular and the main diagonal elements are equal to one, the factor before the determinant corresponds to the value of the determinant itself.

$\mathrm{det\; A}=\left|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ {a}_{j1}& \phantom{\rule{0.3em}{0ex}}{a}_{j2}& \dots & \phantom{\rule{0.3em}{0ex}}{a}_{jn}\\ & \vdots \\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nn}\end{array}\right|={\lambda}\phantom{\rule{0.3em}{0ex}}\left|\begin{array}{cccc}1& {a}_{12}& \dots & {a}_{1n}\\ 0& 1& \dots & {a}_{jn}\\ & \vdots \\ 0& 0& \dots & 1\end{array}\right|={\lambda}\phantom{\rule{0.3em}{0ex}}\mathrm{det\; A\text{'}}={\lambda}$

Here is a list of of further useful calculators:

Index Matrix Determinant Determinant 2x2 Determinant 3x3 Determinant 3x3 symbolic Determinant 4x4 Determinant 4x4 symbolic Determinant 5x5 Determinant 5x5 symbolic Determinant NxN