# Determinant by Gaussian elimination

Finding matrix determinant using Gaussian elimination in complex, rational or real numbers.

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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(n^{3}) operations. It uses Gaussian elimination to convert an input matrix to the triangular form. The calculator supports rational and complex numbers.

### 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).

#### Similar calculators

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**PLANETCALC, Determinant by Gaussian elimination

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