Determinant calculator
Compute the determinant of any square matrix up to 50×50 and see every step — by Gaussian elimination, cofactor expansion, the rule of Sarrus or Montante's method.
Type your matrix above, open the Advanced tab and press det(A). The answer appears immediately; opening the step-by-step panel shows how it was reached, and lets you switch between four different methods for the same matrix — useful when your class was taught one of them in particular.
What the determinant tells you
It is a single number, defined only for square matrices, and its most important job is answering one question: is this matrix invertible? If the determinant is zero the matrix is singular — its rows are linearly dependent, it has no inverse, and a system built on it has either no solution or infinitely many. Any non-zero value means the matrix is invertible.
Geometrically it is the factor by which the transformation scales area in 2D or volume in 3D. A determinant of −2 doubles areas and flips orientation; a determinant of 0 flattens space onto a line or a point, which is precisely why the operation cannot be undone.
Worked example
For a 3×3 matrix, the rule of Sarrus adds the three down-right products and subtracts the three down-left ones:
| 2 | 1 | 3 |
| 0 | 4 | 1 |
| 5 | 2 | 0 |
The calculator uses the closed formula for 2×2 and 3×3, then switches to LU factorisation with partial pivoting from 4×4 upward — cofactor expansion costs n! operations and becomes unusable well before 50×50.
Common mistakes
- Using Sarrus on a 4×4. There is no 4×4 version of the rule. Use elimination or cofactor expansion.
- Reading the bars as absolute value.
|A|is standard notation for the determinant, and it is very often negative. - Forgetting the sign on a row swap. Every swap during elimination multiplies the determinant by −1.
- Expecting
det(A + B) = det(A) + det(B). That is false. Determinants multiply, they do not add.