The product $AB$ is defined only when the number of columns in $A$ equals the number of rows in $B$.
$$A_{m \times k} \times B_{k \times n} = C_{m \times n}$$
Each entry $c_{ij}$ in the product $C$ is the dot product of row $i$ of $A$ and column $j$ of $B$:
$$c_{ij} = \sum_{r=1}^{k} a_{ir} \cdot b_{rj}$$
$$A = \begin{pmatrix} 3 & 1 \ 2 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 0 \ 2 & 3 \end{pmatrix}$$
$$AB = \begin{pmatrix} 3(5)+1(2) & 3(0)+1(3) \ 2(5)+4(2) & 2(0)+4(3) \end{pmatrix} = \begin{pmatrix} 17 & 3 \ 18 & 12 \end{pmatrix}$$
A factory makes two products P and Q. Materials required per unit:
$$\text{Materials} = \begin{pmatrix} 4 & 2 \ 3 & 5 \end{pmatrix} \quad (\text{rows: P, Q; columns: steel, plastic})$$
Production quantities (units per week):
$$\text{Units} = \begin{pmatrix} 100 \ 60 \end{pmatrix} \quad (\text{rows: P, Q})$$
Total materials needed:
$$\text{Materials} \times \text{Units} = \begin{pmatrix} 4(100)+2(60) \ 3(100)+5(60) \end{pmatrix} = \begin{pmatrix} 520 \ 600 \end{pmatrix}$$
520 kg of steel and 600 kg of plastic are required.
| Property | Statement |
|---|---|
| Not commutative | $AB \neq BA$ in general |
| Associative | $(AB)C = A(BC)$ |
| Identity | $AI = IA = A$ |
| Zero | $AO = OA = O$ |
| Distributive | $A(B+C) = AB + AC$ |
For a square matrix $A$:
$$A^2 = AA, \quad A^3 = AAA, \quad \text{etc.}$$
Powers are used in transition and Markov chain problems to find state distributions after $n$ steps.
COMMON MISTAKE: Reversing the order of multiplication. If $A$ is \$2 \times 3$ and $B$ is \$3 \times 4$, then $AB$ is valid but $BA$ requires $B$ to be $m \times 2$, which it is not here.
EXAM TIP: Write out the orders of each matrix before multiplying. Then confirm the inner dimensions match and record the outer dimensions for the result order.
