Matrix Determinant Calculator
A Single Number That Summarizes a Matrix
The determinant compresses an entire square matrix into one scalar value that tells you something fundamental about the transformation it represents — whether it can be inverted, and how much it scales area or volume. A determinant of zero means the matrix is singular: it collapses space down a dimension and has no inverse.
The Formulas
2×2: det(A) = a11×a22 − a12×a21
3×3 (cofactor expansion along row 1):
det(A) = a11(a22a33−a23a32) − a12(a21a33−a23a31) + a13(a21a32−a22a31)
3×3 (cofactor expansion along row 1):
det(A) = a11(a22a33−a23a32) − a12(a21a33−a23a31) + a13(a21a32−a22a31)
Worked Examples
| Matrix | Determinant |
|---|---|
| [4, 7; 2, 6] | (4×6) − (7×2) = 10 |
| [1,2,3; 0,1,4; 5,6,0] | 1 |
Where This Matters
- Solvability of linear systems — a non-zero coefficient-matrix determinant guarantees a system of equations has exactly one solution.
- Area and volume scaling — the absolute value of a 2×2 or 3×3 determinant gives the area or volume scale factor of the linear transformation the matrix represents.
- Testing invertibility — before attempting to invert a matrix, checking its determinant is non-zero avoids wasted computation on a singular matrix.
How to Use This Calculator
- Choose the matrix size: 2 x 2 Matrix or 3 x 3 Matrix.
- Enter each cell value (row1 col1, row1 col2, and so on).
- Select Calculate to get the determinant.
Related Calculations
A non-zero determinant means you can invert the matrix with the Matrix Inverse Calculator, or use it directly in the Cramer's Rule Calculator.