Matrix multiplication

Matrix multiplication composes linear maps. Each result entry is a row-column dot product, and the inner dimensions must match.

Ask the Linear algebra assistant 1 min read · Updated September 9, 2026

Definition

Matrix multiplication composes linear maps. Each result entry is a row-column dot product, and the inner dimensions must match.

Applying B and then A corresponds to AB, so the order is significant.

If A is 2 by 3 and B is 3 by 4, AB is 2 by 4. BA is not defined for those shapes.

For $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times p}$ , $(AB)_{ij}=\sum_k A_{ik}B_{kj}$ .

Check tensor shapes before optimizing a neural layer; transposition mistakes can produce plausible but incorrect values.