Mathematics · Ch 3 — Matrices

Properties of Multiplication of Matrices

3.4.6

Properties of Multiplication of Matrices

Matrix multiplication follows algebraic rules similar to — but not identical with — those for real numbers. The key difference is that it is not commutative: in general, $AB \neq BA$ . However, three fundamental properties do hold whenever the products are defined.

1. The Associative Law

A

The grouping of the factors does not affect the result: you may multiply $AB$ first, then by $C$ , or $BC$ first, then pre-multiply by $A$ .

Note

The associative law is what lets us write $ABC$ without parentheses — the product is unambiguous.

2. The Distributive Laws

A

Matrix multiplication distributes over addition, in both a left and a right form.

Watch out

Order matters. Since matrix multiplication is not commutative, $A(B+C) = AB + AC$ is true, but $(B+C)A = BA + CA$ is a different (equally true) statement. You cannot swap the order of factors when distributing.

3. The Existence of Multiplicative Identity

A

The identity matrix $I$ (1's on the main diagonal, 0's elsewhere) acts like the number 1: multiplying any matrix by the identity of compatible order leaves it unchanged.

Note

For non-square matrices the identity must match the dimensions. If $A$ is $m \times n$ , then $I_m A = A$ and $A I_n = A$ , with $I_m$ and $I_n$ being different-sized identity matrices.

4. Application: Polynomials in Matrices

The properties above let us work with matrix polynomials. For a square matrix $A$ ,

A^2 = A \cdot A, \quad A^3 = A^2 \cdot A, \quad \text{and so on}

with $A^0 = I$ by convention. …

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