Matrix Multiplication: A Step-by-Step Guide With Worked Examples

Matrix multiplication is the operation that drives linear algebra, computer graphics, machine learning, and physics simulations. Yet most students learn it as a mechanical recipe and never see why it is defined the way it is. This guide gives you both the recipe and the intuition.

Dimension rule first

Before you compute anything, check dimensions. To multiply $A \cdot B$:

• $A$ must have shape $m \times n$
• $B$ must have shape $n \times p$
• Result $AB$ has shape $m \times p$

The inner dimensions must match ($n = n$); the outer dimensions become the result shape.

If you ever try to multiply a $3 \times 4$ by a $5 \times 2$, the operation is undefined — no amount of arithmetic will save you.

The row-times-column recipe

The $(i, j)$ entry of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$:

$(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$

Worked example

$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$

Compute $AB$:

• $(AB)_{11} = 1\cdot 5 + 2\cdot 7 = 19$
• $(AB)_{12} = 1\cdot 6 + 2\cdot 8 = 22$
• $(AB)_{21} = 3\cdot 5 + 4\cdot 7 = 43$
• $(AB)_{22} = 3\cdot 6 + 4\cdot 8 = 50$

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

Why is multiplication defined this way?

Matrices represent linear maps between vector spaces. If $A$ maps from $\mathbb{R}^n$ to $\mathbb{R}^m$, and $B$ maps from $\mathbb{R}^p$ to $\mathbb{R}^n$, then $AB$ should be the composition of those maps. The row-times-column rule is precisely what produces composition. The recipe is not arbitrary — it falls out of the requirement that $AB$ encode "first apply $B$, then apply $A$".

Properties (and non-properties!)

Property	Holds?
$A(BC) = (AB)C$ associative	Yes
$A(B + C) = AB + AC$ distributive	Yes
$AB = BA$ commutative	No, in general
$AB = 0 \Rightarrow A = 0$ or $B = 0$	No

The non-commutativity is the single biggest mental adjustment from scalar arithmetic.

Common mistakes

• Adding instead of multiplying the row-column products (you do both — multiply pairwise then sum).
• Switching the dimension check order — it must be $(m \times n)(n \times p)$, not $(n \times m)(n \times p)$.
• Assuming commutativity — $AB$ may not even be defined if $BA$ is.

Try with the AI Matrix Solver

Type any pair of matrices into the Matrix Calculator for fully-shown row-by-row work.

Related references:

• Determinant Calculator — pairs naturally with $2\times 2$ products
• Inverse Calculator — uses $AB = I$ as the defining relation
• Vector Calculator — dot product underlies every entry