Matrix Multiplication: Rows Times Columns

The rule

For $A$ of size $m \times n$ and $B$ of size $n \times p$, the product $AB$ has size $m \times p$, with entries $(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$

In words: $(AB)_{ij}$ is the dot product of row $i$ of $A$ with column $j$ of $B$.

The "inner dimension" $n$ must match — otherwise the product is undefined.

Step-by-step

Setup: $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. Both $2 \times 2$, product is $2 \times 2$.

Compute each entry:

$(AB)_{11}$ = row 1 of $A$ · column 1 of $B$: $(1)(5) + (2)(7) = 5 + 14 = 19$

$(AB)_{12}$ = row 1 of $A$ · column 2 of $B$: $(1)(6) + (2)(8) = 6 + 16 = 22$

$(AB)_{21}$ = row 2 of $A$ · column 1 of $B$: $(3)(5) + (4)(7) = 15 + 28 = 43$

$(AB)_{22}$ = row 2 of $A$ · column 2 of $B$: $(3)(6) + (4)(8) = 18 + 32 = 50$

$\boxed{AB = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}}$

Matrix multiplication is NOT commutative

$BA = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 23 & 34 \\ 31 & 46 \end{pmatrix} \ne AB$

In general $AB \ne BA$. They can even have different shapes (e.g. $A$ is $2 \times 3$, $B$ is $3 \times 2$: $AB$ is $2 \times 2$, but $BA$ is $3 \times 3$).

Properties (what IS true)

• Associative: $A(BC) = (AB)C$.
• Distributive: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$.
• Identity: $AI = A = IA$ with $I$ the appropriately-sized identity matrix.
• Transpose swap: $(AB)^T = B^T A^T$.

Geometric interpretation

Matrix $A$ represents a linear transformation. The product $AB$ represents the composition — first apply $B$, then apply $A$. This naturally explains why matrix multiplication is associative (composition is) but not commutative (order of transformations matters).

Common mistakes

• Row-times-row or column-times-column. The rule is row (of $A$) times column (of $B$), period.
• Assuming $AB = BA$. Matrix multiplication rarely commutes.
• Ignoring shape mismatches. A $2 \times 3$ times a $2 \times 3$ is not defined — inner dimensions ($3$ and $2$) don't match.
• Claiming $AB = 0 \implies A = 0$ or $B = 0$. False in general: two nonzero matrices can multiply to zero (they're "zero divisors").

Try it in the visualization

The row of $A$ and the column of $B$ highlight as each entry of $AB$ is computed. Their element-wise products add up live to produce each output entry.