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Algebraic Properties of Matrix Multiplication

Matrix multiplication satisfies several familiar algebraic properties, but with important differences from scalar multiplication.

Works: Associativity: $(AB)C = A(BC)$ . Distributivity: $A(B+C) = AB + AC$ and $(B+C)A = BA + CA$ . Identity: $I_mA = A = AI_n$ . Scalar: $r(AB) = (rA)B = A(rB)$ .

Does NOT work in general: Commutativity ( $AB \neq BA$ in general), cancellation ( $AB = AC$ does not imply $B = C$ ), zero products ( $AB = 0$ does not imply $A = 0$ or $B = 0$ ).

Formal View

Theorem 4.5 — Properties of Matrix Multiplication

Let

A, B, C

be matrices of compatible dimensions and

r \in \mathbb{R}

: \begin{enumerate} \item

(AB)C = A(BC)

(associativity) \item

A(B + C) = AB + AC

(left distributivity) \item

(B + C)A = BA + CA

(right distributivity) \item

I_m A = A

and

A I_n = A

(identity) \item

r(AB) = (rA)B = A(rB)

(scalar compatibility) \end{enumerate}

Why This Matters

These properties justify algebraic manipulation of matrix expressions.

Associativity allows us to evaluate $(AB)C$ or $A(BC)$ — whichever is cheaper
Distributivity enables expanding matrix polynomials like $(A+B)^2 = A^2 + AB + BA + B^2$ (note: $AB \neq BA$ !)

Learning Resources

Properties of matrix multiplication

Khan Academy

Associativity, distributivity, and the identity matrix.

11 min

Quiz

Question 1

For all matrices $A$ and $B$ , $(A+B)^2 = A^2 + 2AB + B^2$ .

Common Mistakes

Treating matrix algebra like scalar algebra — especially the commutativity assumption.
Thinking $(AB)^T = A^T B^T$ — the correct formula is $(AB)^T = B^T A^T$ (order reverses).