Matrix Determinant Calculator
Compute the determinant of a square matrix by Gaussian elimination with partial pivoting.
Description
Calculate the determinant of a square matrix with pivoted Gaussian elimination.
A matrix determinant is a single number that summarizes important properties of a square matrix. It indicates whether the matrix is invertible, gives the signed area or volume scale of its linear transformation, and appears in systems of equations, eigenvalue calculations, and change-of-variables formulas.
When to use Matrix Determinant Calculator
- Check whether a square matrix is singular
- Measure signed area or volume scaling
- Verify a hand calculation for a small determinant
How the calculation works
The calculator uses Gaussian elimination with partial pivoting. Row swaps change the determinant's sign, elimination preserves it, and the final determinant is the signed product of the diagonal pivots. This approach avoids the factorial work of a cofactor expansion.
det(A) = sign(row swaps) × product of elimination pivots Interpreting the result
A nonzero determinant means the matrix has an inverse. Zero means its rows or columns are linearly dependent. Sign indicates orientation reversal, while absolute value describes the transformation's area or volume scale; it is not a general measure of matrix size.
Important limitations
- Only non-empty square matrices have a determinant.
- Floating-point elimination can make a mathematically tiny determinant sensitive to rounding.
- A determinant alone does not show which rows are dependent or how ill-conditioned a matrix is.