Mathematics · Ch 4 — Determinants and Matrices

Properties of Matrix Multiplication

4.6

Properties of Matrix Multiplication

4.6 Properties of Matrix Multiplication

Not commutative. In general $AB\ne BA$ for matrices $A,B$ (§4.5.4 already showed this).
Associative. $(AB)C=A(BC)$ whenever the orders are suitable for multiplication.
Distributive over addition. $A(B+C)=AB+AC$ (left distributive law) and $(B+C)A=BA+CA$ (right distributive law).
Multiplicative identity. For a square matrix $A$ , the identity matrix $I$ of the same order satisfies $AI=IA=A$ .
Null-matrix absorption. For any matrix $A$ , there is a null matrix $O$ (conformable) with $AO=O$ and $OA=O$ .
Zero product without a zero factor. The product of two non-zero matrices can itself be the zero matrix: $AB=O$ is possible even though $A\ne O$ and $B\ne O$ — a genuine departure from ordinary number arithmetic, where $ab=0\Rightarrow a=0$ or $b=0$ .
Positive integer powers. For a square matrix $A$ , $A^2=AA,\ A^3=AAA,\dots,A^n=\underbrace{AA\cdots A}_{n\text{ times}}$ .

Consequences of the distributive law (for square $A,B$ of the same order)

Because $AB\ne BA$ in general, the familiar numeric expansions pick up extra cross-terms that must be kept separate:

(A+B)^2=A^2+AB+BA+B^2,\qquad (A-B)^2=A^2-AB-BA+B^2,\qquad (A+B)(A-B)=A^2+BA-AB-B^2

(Only when $AB=BA$ do these collapse to the familiar $A^2\pm2AB+B^2$ and $A^2-B^2$ forms.)

Worked Examples

Example 1. Show $AB$ is non-singular for suitable matrices $A,B$ .

Step 1: Compute $AB$ directly, entry by entry, using the row-times-column rule.

Step 2: Compute $|AB|$ by cofactor expansion.

Step 3: Since $|AB|\ne0$ , $AB$ is non-singular by definition (§4.4.1, type 14).

Example 2. If $A=\begin{bmatrix}1&3&3\\3&1&3\\3&3&1\end{bmatrix}$ , prove $A^2-5A$ is a scalar matrix.

Step 1: $A^2=AA$ ; computing entry-by-entry (each diagonal entry of $A^2$ is $1+9+9=19$ , each off-diagonal entry is $3+3+9=15$ ), so $A^2=\begin{bmatrix}19&15&15\\15&19&15\\15&15&19\end{bmatrix}$ .

Step 2: $5A=\begin{bmatrix}5&15&15\\15&5&15\\15&15&5\end{bmatrix}$ . …

Misc 4.6Illustration of Property 2 (associativity) with three numeric matrices

Worked out. Student counts per game per school are arranged in one matrix and the per-student coaching/equipment fee per game in another; multiplying the two matrices gives each school's total coaching and equipment expense, worked out school by school. …

Misc 4.6Illustration of Property 4 (identity matrix) with a numeric matrix

Misc 4.6Illustration of Property 6 (zero product from nonzero factors)

Misc 4.6Activity — computing AB−2I for two given 2×2 matrices

Misc 4.6Solved Example 1 — showing a product matrix AB is non-singular

Misc 4.6Solved Example 2 — proving a combination A²−5A is a scalar matrix

Misc 4.6Solved Example 3 — finding an unknown k from a matrix polynomial equation

Misc 4.6Solved Example 4 — finding unknowns x,y from a chained matrix product

Misc 4.6Solved Example 5 — finding θ from a matrix-product-equals-scalar equation

Misc 4.6Remark — expanding (A±B)² and (A+B)(A−B) for matrices

Misc 4.6Solved Example 6 — proving a matrix-power formula by mathematical induction

Misc 4.6Solved Example 7 — word problem (bookseller's total receipt via matrix multiplication)

Misc 4.6Solved Example 8 — word problem with two data tables multiplied (sports academy fees)

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Properties of Matrix Multiplication