Matrix Determinant Calculator
Find the determinant of a square matrix. Supports 1×1 through 6×6 matrices. The determinant is a scalar that encodes important geometric and algebraic properties.
Notes
What is the Determinant?
The determinant is a scalar value associated with a square matrix. Geometrically it is the signed volume scaling factor: if det(A) > 0 the transformation preserves orientation; if det(A) < 0 it reverses it. If det(A) = 0, the matrix collapses space to a lower dimension and has no inverse.
2×2 Determinant
Example — compute det([[3,8],[4,6]]):
| 3 | 8 |
| 4 | 6 |
det = (3)(6) − (8)(4) = 18 − 32 = −14
3×3 Determinant — Laplace Expansion Along Row 1
Example — compute det of A = [[2,3,1],[0,4,5],[1,2,3]]:
| 2 | 3 | 1 |
| 0 | 4 | 5 |
| 1 | 2 | 3 |
Minor M₁₁ (delete row 1, col 1):
| 4 | 5 |
| 2 | 3 |
M₁₁ = 4·3 − 5·2 = 12 − 10 = 2
Minor M₁₂ (delete row 1, col 2):
| 0 | 5 |
| 1 | 3 |
M₁₂ = 0·3 − 5·1 = −5
Minor M₁₃ (delete row 1, col 3):
| 0 | 4 |
| 1 | 2 |
M₁₃ = 0·2 − 4·1 = −4
Sarrus' Rule (3×3 only)
Key Properties
| Property | Rule |
|---|---|
| Product | det(AB) = det(A) · det(B) |
| Transpose | det(A^T) = det(A) |
| Scalar | det(kA) = k^n · det(A) |
| Row swap | Swapping rows negates det |
| Triangular | det = product of diagonal entries |
| Singular | det = 0 → no inverse |
- Matrix Calculations Guide — Matrix Determinant — In-depth notes on Matrix Determinant with worked examples
Frequently Asked Questions
What does a zero determinant mean?
A zero determinant means the matrix is singular — it has no inverse, and its rows (or columns) are linearly dependent.
How does the determinant change with row operations?
Swapping two rows negates the determinant. Multiplying a row by k scales the determinant by k. Adding a multiple of one row to another leaves the determinant unchanged.
What is det(AB)?
det(AB) = det(A) · det(B) — the determinant is multiplicative.
How do I compute the determinant of a 4×4 matrix?
Expand along any row or column using cofactors, each requiring a 3×3 determinant. Alternatively, row-reduce to upper triangular form (track sign changes from swaps) and multiply the diagonal.