Matrix Multiplication is not Commutative

Theorem
Let $R$ be a ring with unity.

Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n \ne 1$.

Let $\map {\MM_R} n$ denote the $n \times n$ matrix space over $R$.

Then (conventional) matrix multiplication over $\map {\MM_R} n$ is not commutative:
 * $\exists \mathbf A, \mathbf B \in \map {\MM_R} n: \mathbf {A B} \ne \mathbf {B A}$

If $R$ is specifically not commutative, then the result holds when $n = 1$ as well.

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:
 * $\exists \mathbf A, \mathbf B \in \map {\MM_R} n: \mathbf {A B} \ne \mathbf {B A}$

$n = 1$
Consider the case where $n = 1$.

Then:

and it follows that (conventional) matrix multiplication over $\map {\MM_R} 1$ is commutative $R$ is a commutative ring.

$R$ not a Ring with Unity
Consider the case where $R$ is not a ring with unity, and is a general ring.

Let $R$ be the trivial ring.

From Matrix Multiplication on Square Matrices over Trivial Ring is Commutative:


 * $\forall \mathbf A, \mathbf B \in \map {\MM_R} n: \mathbf {A B} = \mathbf {B A}$

Hence the result does not follow for all rings.

It is not established at this point on exactly which rings (conventional) matrix multiplication $\map {\MM_R} n$ commutes.

However, the existence of just one such ring (the trivial ring) warns us that we cannot apply the main result to all rings.

Matrices are not Square
We note that $\mathbf A \mathbf B$ is defined when:
 * $\mathbf A = \sqbrk a_{m n}$ is an $m \times n$ matrix
 * $\mathbf B = \sqbrk b_{n p}$ is an $n \times p$ matrix.

Hence for both $\mathbf A \mathbf B$ and $\mathbf B \mathbf A$ to be defined, it is necessary that:
 * $\mathbf A = \sqbrk a_{m n}$ is an $m \times n$ matrix
 * $\mathbf B = \sqbrk b_{n p}$ is an $n \times m$ matrix

for some $m, n \in \Z_{>0}$.

But in this situation:
 * $\mathbf A \mathbf B$ is an $m \times m$ matrix

while:
 * $\mathbf B \mathbf A$ is an $n \times n$ matrix

and so if $\mathbf A$ and $\mathbf B$ are not square matrices, they cannot commute.

Basis for the Induction
From Matrix Multiplication is not Commutative/Order 2 Square Matrices, it is seen that there exist $2 \times 2$ matrices that don't commute under matrix multiplication, thus proving the result for $n = 2$.

$\map P 2$ is the case:
 * $\exists \mathbf A, \mathbf B \in \map {\MM_R} 2: \mathbf {A B} \ne \mathbf {B A}$

This is demonstrated in Matrix Multiplication is not Commutative: Order $2$ Square Matrices.

Thus $\map P 2$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * $\exists \mathbf A, \mathbf B \in \map {\MM_R} k: \mathbf {A B} \ne \mathbf {B A}$

from which it is to be shown that:
 * $\exists \mathbf A, \mathbf B \in \map {\MM_R} {k + 1}: \mathbf {A B} \ne \mathbf {B A}$

Induction Step
This is the induction step:

From the induction hypothesis, it is assumed that there exist $2$ order $k$ square matrices $\mathbf A$ and $\mathbf B$ such that $\mathbf {A B} \ne \mathbf {B A}$.

For an order $n$ square matrix $\mathbf D$, let $\mathbf {D'}$ be the square matrix of order $n + 1$ defined as:


 * $d'_{i j} = \begin {cases} d_{i j} & : i < n + 1 \land j < n + 1 \\ 0 & : i = n + 1 \lor j = n + 1 \end{cases}$

Thus $\mathbf D'$ is just $\mathbf D$ with a zero row and zero column added at the ends.

We have that $\mathbf D$ is a submatrix of $\mathbf D'$.

Now:


 * $\paren {a' b'}_{i j} = \begin{cases} \displaystyle \sum_{r \mathop = 1}^{n + 1} \mathbf a'_{i r} b'_{r j} & : i < n + 1 \land j < n + 1 \\ 0 & : i = n + 1 \lor j = n + 1 \end{cases}$

But:

and so:

Thus it is seen that:


 * $\exists \mathbf A', \mathbf B' \in \MM_{n + 1 \times n + 1}: \mathbf A' \mathbf B' \ne \mathbf B' \mathbf A'$

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\exists \mathbf A, \mathbf B \in \map {\MM_R} n: \mathbf {A B} \ne \mathbf {B A}$

and by definition (conventional) matrix multiplication over $\map {\MM_R} n$ is not commutative.