Matrix Multiplication is not Commutative

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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:

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


Edge Cases

$n = 1$

Consider the case where $n = 1$.

Then:

\(\ds \mathbf {A B}\) \(=\) \(\ds a_{11} b_{11}\)
\(\ds \mathbf {B A}\) \(=\) \(\ds b_{11} a_{11}\)


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

$\map P 2$ 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:

\(\ds \sum_{r \mathop = 1}^{n + 1} a'_{i r} b'_{r j}\) \(=\) \(\ds a'_{i \paren {n + 1} } b'_{\paren {n + 1} i} + \sum_{r \mathop = 1}^n a'_{i r} b'_{r j}\)
\(\ds \) \(=\) \(\ds \sum_{r \mathop = 1}^n a_{i r} b_{r j}\)


and so:

\(\ds \mathbf A' \mathbf B' \paren {n + 1, n + 1}\) \(=\) \(\ds \paren {\mathbf {A B} }' \paren {n + 1, n + 1}\)
\(\ds \) \(=\) \(\ds \mathbf {A B}\)
\(\ds \) \(\ne\) \(\ds \mathbf {B A}\)
\(\ds \) \(=\) \(\ds \paren {\mathbf {B A} }' \paren {n + 1, n + 1}\)
\(\ds \) \(=\) \(\ds \mathbf B' \mathbf A' \paren {n + 1; n + 1}\)


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.

$\blacksquare$


Examples

Arbitrary $2 \times 2$ Matrices

Consider the matrices:

\(\ds \mathbf A\) \(=\) \(\ds \begin {pmatrix} 1 & 2 \\ -1 & 0 \end {pmatrix}\)
\(\ds \mathbf B\) \(=\) \(\ds \begin {pmatrix} 1 & -1 \\ 0 & 1 \end {pmatrix}\)

We have:

\(\ds \mathbf A \mathbf B\) \(=\) \(\ds \begin {pmatrix} 1 & 1 \\ -1 & 1 \end {pmatrix}\)
\(\ds \mathbf B \mathbf A\) \(=\) \(\ds \begin {pmatrix} 2 & 2 \\ -1 & 0 \end {pmatrix}\)

and it is seen that $\mathbf A \mathbf B \ne \mathbf B \mathbf A$.


Also see


Sources