Matrix Product

Matrix multiplication is the algebra of chained dependencies. It is not entrywise; it works row-by-column.

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

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

Dimension rule: $A_{m \times n} \cdot B_{n \times p} = C_{m \times p}$ . Inner dimensions must match and disappear.

Action	Rule	Result
Inner Check	$(m \times \mathbf{n})(\mathbf{n} \times p)$	Must be equal
Outer Shape	$(\mathbf{m} \times n)(n \times \mathbf{p})$	Final size $\mathbf{m \times p}$

Tip: Always write dimensions side by side. Ten seconds of checking catches most errors.

A chain links only where the hand-off dimension matches -- and once it

Why business loves it: If $Q$ holds units sold and $V$ holds prices and costs, $VQ$ delivers revenue and cost per branch in one stroke.

The identity matrix $I$ has ones on the diagonal and zeros elsewhere: $AI = IA = A$ .

Common pitfall: Assuming $AB = BA$ . Matrix multiplication chains processes, so order is always critical.

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