Most students, on first encountering the definition of matrix multiplication, have the same reaction: why on earth is it defined this way?

Given $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ and $B=\left(\begin{array}{cc}e& f\\ g& h\end{array}\right)$, the product is

Multiplying entry-by-entry would be much simpler. But the "row-times-column" rule is not a convention chosen for aesthetic reasons. It is forced on us by the fact that matrices represent linear maps, and matrix multiplication represents composition.

1 Matrices as linear maps

A linear map $T:{\mathbb{R}}^2\to {\mathbb{R}}^3$ is completely determined by what it does to a basis. If we know $T(e_1)$ and $T(e_2)$, we know everything.

The recipe is simple: place the images of the basis vectors as columns. The resulting matrix is the representation of $T$.

2 Composition forces the rule

Here is the crux. Take two linear maps and compose them — apply one after the other — and compute the representation matrix of the composition by hand.

Now look carefully at what happened. To get the $(i,j)$ entry of the result, we took the dot product of the $i$-th row of $B$ with the $j$-th column of $A$. That is precisely the definition of matrix multiplication:

A perfect match.

3 Size constraints become obvious

An $m\times n$ matrix represents a map ${\mathbb{R}}^n\to {\mathbb{R}}^m$, and a $p\times q$ matrix represents a map ${\mathbb{R}}^q\to {\mathbb{R}}^p$. The composition $A\circ B$ makes sense only when the output of $B$ can be fed into $A$ — that is, when $p=n$. The result is a map ${\mathbb{R}}^q\to {\mathbb{R}}^m$, hence an $m\times q$ matrix.

4 Associativity is just composition

The associative law $(AB)C=A(BC)$ for matrices is a reflection of the associativity of function composition: $(f\circ g)\circ h=f\circ (g\circ h)$. Applying three maps in succession, it does not matter which pair you compose first — the final result is the same. There is nothing to prove about matrices specifically; associativity is inherited from the nature of function composition itself.

5 Why$AB\ne BA$

The non-commutativity of matrix multiplication is also transparent from the map perspective.

6 Change of basis: the same map in a different outfit

The same linear map can look very different depending on the choice of basis. If $P$ is the change-of-basis matrix from basis $B$ to basis $B^{\prime }$, then the representation matrix transforms as

7 The takeaway

The "row-times-column" rule for matrix multiplication is not an arbitrary convention. It is the unique rule that makes the product of two matrices equal the matrix of the composed maps. Size constraints, associativity, non-commutativity — they all follow from thinking of matrices not as tables of numbers but as proxies for linear maps.