Matrix Determinant Calculator
Compute the determinant of any square matrix using LU factorisation with partial pivoting.
Inputs
Determinant
-14.00000000
Singular?
No
Matrix Size
2×2
Step by step
Matrix A
= 3 8 4 6
Method
= LU decomposition with partial pivoting
det(A) = sign × product of U diagonal
= -14
Singular?
= No — the matrix is invertible
How it works
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
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.