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Determinants

Matrix determinant calculator. Enter a square matrix of any order to compute det(A) with expansion steps—for linear algebra homework checks.

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det(A) = Σ (−1)^τ a₁p₁·…·aₙpₙ
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Determinants

1. Definition
The determinant of an n×n matrix A is a number, det(A) or |A|, the signed sum of n! products.
2. Symbols
SymbolMeaning
det(A)determinant of A
τpermutation inversion count
main diagonalproduct for upper triangular
3. How this tool works
  • 2×2: det = ad−bc;
  • Upper triangular determinant = product of diagonal entries;
  • Swapping two rows flips the sign; scaling a row by k scales det by k.
4. Steps
  1. Enter A;
  2. Click Calculate to see det(A);
  3. det=0 means the matrix is singular.
5. Example
Example: A=6 1 1;4 −2 5;2 8 7. By elimination det=−306.
6. Pitfalls
Only square matrices have a determinant;
det=0 means no inverse;
A determinant is a number, a matrix is an array.

❓ FAQ｜Determinants

2×2 determinant formula?
ad−bc.
What does det=0 mean?
The matrix is singular, not invertible, with linearly dependent rows/columns.
Properties?
Row swap flips sign; scaling a row by k scales det; det(AB)=detA·detB.
Does transposing change the determinant?
No: det(Aᵀ)=det(A).
Can a non-square matrix have a determinant?
No.