Matrix Determinant Calculator
Compute the determinant of any square matrix using LU factorisation with partial pivoting.
About Matrix Determinant Calculator
The determinant of a square matrix is computed via LU decomposition with partial pivoting. det(A) = (sign from row swaps) × product of diagonal entries of U. This is O(n³) and numerically stable, unlike cofactor expansion which is O(n!).
Formula
Via LU Decomposition
det(A) = (−1)^s × ∏ Uᵢᵢ where s = number of row swaps
- U
- Upper triangular factor
- s
- Number of pivot row swaps
Worked example
With the default values below, Matrix Determinant Calculator returns Determinant of -14.00000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Matrix A (rows separated by newlines or semicolons; values by commas or spaces)
- 3 8 4 6
Results
- Determinant
- -14.00000000
- Singular?
- No
- Matrix Size
- 2×2
Step by step
- 1Matrix A= 3 8 4 6
- 2Method= LU decomposition with partial pivoting
- 3det(A) = sign × product of U diagonal= -14
- 4Singular?= No — the matrix is invertible
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What does the determinant tell us?
A non-zero determinant means the matrix is invertible. Geometrically, |det(A)| is the volume scaling factor of the linear transformation represented by A.
Why not use cofactor expansion?
Cofactor expansion has O(n!) complexity — impractical for n > 5. LU decomposition computes the determinant in O(n³) time and handles numerical issues via partial pivoting.