Matrix Determinant

Determinant Properties

det(I) = 1

det(I) = 1 — The identity matrix always has determinant 1.

det(AB) = det(A) · det(B) — Determinant is multiplicative over matrix products.

det(Aᵀ) = det(A) — Transposing a matrix does not change its determinant.

Swap rows → negate det — Each row swap multiplies the determinant by −1.

Two equal rows → det = 0 — Linearly dependent rows force the determinant to zero.

Scale one row by k → det scales by k — Only that one row's contribution is scaled.

For a 2×2 matrix: det([[a,b],[c,d]]) = ad − bc. For 3×3, use cofactor expansion along any row or column — the value is always the same.

Geometric Meaning

|det| = area / volume

2×2 matrix: |det(A)| equals the area of the parallelogram formed by the two column vectors [a, c] and [b, d]. If det > 0, the orientation is preserved; if det < 0, it is reversed.

3×3 matrix: |det(A)| equals the volume of the parallelepiped formed by the three column vectors. Again, sign indicates orientation.

det = 0: The columns are linearly dependent — they lie in a lower-dimensional space (a line for 2×2, a plane for 3×3), so there is no area or volume. The matrix is singular and has no inverse.

A matrix is invertible if and only if its determinant is nonzero.
det(A⁻¹) = 1 / det(A) when A is invertible.
Cramer's Rule uses determinants to solve linear systems.
The sign of det tells you whether the transformation flips orientation.

Determinants still tricky?