Properties of matrix multiplication

Matrix multiplication is distributive, A(B + C) = AB + AC, and associative, A(BC) = (AB)C.

It is not commutative: AB = BA does not hold in general, unlike multiplication of numbers. Often BA is not even defined, or has a different shape. If A is 2 × 3 and B is 3 × 2, AB is 2 × 2 but BA is 3 × 3.

A whole system of linear equations can be written as one product, Ax = b. A ∈ ℝᵐˣⁿ holds the known coefficients, b ∈ ℝᵐ the known right-hand sides, and x ∈ ℝⁿ the unknowns. Each row of A, together with one element of b, is one equation.