Matrix Product

Row Meets Column

Matrix multiplication is not entrywise — it is the algebra of chained dependencies, and it works row-by-column:

$$(AB)_{ij} = \sum_k a_{ik}\,b_{kj}$$

Entry $(i,j)$ of the product is the dot product of row $i$ of $A$ with column $j$ of $B$: multiply matching components, add them up.

Dimension rule: $A_{m\times n} \cdot B_{n \times p} = C_{m\times p}$ — the inner dimensions must match and then disappear:

$$(m \times \underline{n})\,(\underline{n} \times p) \;\rightarrow\; m \times p$$

Why business loves it. Suppose $Q$ holds units sold (products by branches) and each product has a price and a unit cost in matrix $V$ (values by products). Then $VQ$ delivers revenue and cost per branch in one stroke. Every 'multiply quantities by rates, then total' calculation — payroll, input requirements, portfolio returns — is a matrix product.

The famous warning: order matters.

$$AB \ne BA \quad \text{(in general)}$$

$BA$ may have different dimensions, or not exist at all, or exist and simply differ. Chains read like processes: raw materials → components → products composes in one direction only.

Other habits to unlearn: $AB = 0$ does not force $A=0$ or $B=0$; and you cannot cancel — $AB = AC$ does not imply $B = C$ unless $A$ is invertible (a determinant question, next lesson).

The identity matrix $I$ (ones on the diagonal, zeros elsewhere) is the neutral element: $AI = IA = A$ — the matrix world's multiply-by-one.

Tip: Before multiplying anything, write the dimensions side by side: $(m \times n)(n \times p)$. Inner numbers must match; outer numbers announce the answer's shape. Ten seconds of dimension checking catches most errors before they exist.

Common pitfall: Assuming $AB = BA$. Matrix multiplication is order-sensitive because it chains dependencies — applying "prices after quantities" is not "quantities after prices." Commuting matrices are the exception, never the rule.