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In linear algebra, det(AB) = det(A) det(B) expresses the multiplicativity of the determinant. This holds for all square n-by-n matrices A and B over a field F (typically the real or complex numbers). The determinant is a scalar-valued function det: M_n(F) -> F that encodes many properties of a linear transformation.

The identity states that the determinant of a product equals the product of the determinants. Equivalently, det(AB) and det(A)det(B) are the same scalar, and since det(B)det(A) = det(A)det(B), the order of multiplication does not affect the value of the determinant for the product.

Geometric interpretation: The determinant of a matrix represents the factor by which the associated linear transformation scales n-dimensional volume. If B is applied first, volumes are scaled by det(B); then A scales by det(A). Therefore, the composition AB scales volumes by det(A)det(B).

Consequences and related facts: det(I) = 1, where I is the identity matrix, and det(A) = 0 if and only if A is singular (not invertible). The determinant of a transpose equals the determinant: det(A^T) = det(A). The multiplicativity also implies det(AB) = det(BA) if A and B are square and of compatible size, yielding det(AB) = det(A) det(B) = det(B) det(A).

Proof sketch: The determinant is defined to be multiplicative on elementary matrices (row swaps, row scalings, and row additions). Since any invertible matrix can be written as a product of elementary matrices, the property extends to all A and B, with continuity extending it to singular cases.