Determinant

The determinant calculator finds the determinant of a square matrix up to 6×6 by Gaussian elimination (reduction to triangular form) or by cofactor expansion along the first row (up to 4×4). For 2×2 and 3×3 matrices the formulas are shown. The calculation is exact, in fractions.

Formulas for small matrices

|a b; c d| = ad − bc

A 3×3 determinant by the rule of Sarrus: three products along the main diagonal and its parallels with a plus sign, three along the other diagonal with a minus sign.

Gaussian elimination

1. Use row operations to make the entries below the main diagonal zero.
2. Adding a multiple of one row to another does not change the determinant; swapping two rows changes its sign.
3. The determinant of a triangular matrix is the product of its diagonal entries.

Cofactor expansion

det A = a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃ − …

M_1j is a minor: the determinant of the matrix without the first row and the j-th column.

Properties

• det A^T = det A,   det(AB) = det A · det B.
• If the rows are linearly dependent (for example, two rows are proportional), the determinant is zero.
• det A ≠ 0 if and only if the matrix is invertible.