Determinant by Gaussian elimination

We already have the matrix determinant calculator, but it employs a straightforward algorithm with complexity O(n!), which makes it unusable for big matrices. The following calculator performs the same task in O(n³) operations. It uses Gaussian elimination to convert an input matrix to the triangular form. The calculator supports rational and complex numbers.

Determinant by Gaussian elimination

0 0 2 4 5 1 3 5 1 3 0 1 1 4 2 1 5 6 3 0 1 4 2 1 4 2 0 1 9 2 3 1 9 5 7 0

Matrix

Calculation precision

Exact

Rounded

Digits after the decimal point: 2

Calculate

Determinant

 

Input matrix

 

Triangular matrix

 

Number of swaps

 

 Link  Save  Widget

Determinant properties used by the calculator algorithm

• The calculator converts the input matrix to the triangular form to calculate the matrix determinant by multiplying its main diagonal elements.
• The conversion is performed by subtracting one row from another multiplied by a scalar coefficient. This operation doesn't change the determinant.
• The calculator swaps two rows if zero appears in the main diagonal. It counts a number of row swaps to change the determinant sign according to the number of swaps (each swap changes the result sign).
• It stops calculation if a zero column is found since the determinant becomes zero in this case.

The determinant has several other interesting properties not used by this calculator (see Wikipedia).