The determinant tells you whether a matrix is invertible (non-zero), by how much it scales areas and volumes, and whether it flips orientation (negative). It appears in Cramer’s rule, eigenvalues and changes of variables in integrals.
How to use the Matrix Determinant Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
For 2 × 2 the cross-product formula is used; for larger matrices the calculator uses elimination for the value and shows the first-row cofactor expansion.
Formula
2 × 2: |A| = a₁b₂ − b₁a₂
n × n: |A| = Σⱼ (−1)¹+j) a₁ⱼ M₁ⱼ
Worked example
Inputs — Matrix size: 3 × 3; a₁: 4; b₁: 1; c₁: 2; a₂: 1; b₂: 3; c₂: 0; a₃: 2; b₃: 0; c₃: 5.
Results — Determinant |A| = 43; Expansion = Expand along the first row (cofactor expansion):
|A| = 4 × 15 − 1 × 5 + 2 × −6
= 60 − 5 − 12 = 43; Matrix = ⎡ 4 1 2 ⎤
⎢ 1 3 0 ⎥
⎣ 2 0 5 ⎦.
Key terms
- Determinant
- A scalar summarising a square matrix.
Frequently asked questions
What does a zero determinant mean?
The matrix is singular – its columns are dependent and it has no inverse.
What is the geometric meaning?
The factor by which the transformation scales areas (2D) or volumes (3D).
What happens if I swap two rows?
The determinant changes sign.
Is det(AB) = det(A)·det(B)?
Yes.