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Definition of matrix multiplication
If $A$ is an $n\times m$ matrix and $B$ is an $m\times k$ matrix, then the product $AB$ is the $n\times k$ matrix whose $(i,j)$ entry is the row-column product of the $i$th row of $A$ with the $j$th column of $B$. That is: \[ (AB)_{i,j} = \text{row}_i(A)\cdot \text{col}_j(B).\] If $A$ is an $n\times m$ matrix and $B$ is an $\ell\times k$ matrix with $m\ne \ell$, then the matrix product $AB$ is undefined.
Examples
- If $\newcommand{\mat}[1]{\begin{bmatrix}#1\end{bmatrix}} A=\mat{1&0&5\\2&-1&3}$ and $B=\mat{1&2\\3&4\\5&6}$, then $AB=\mat{26&32\\14&18}$ and $BA=\mat{5&-2&11\\11&-4&27\\17&-6&43}$. Note that $AB$ and $BA$ are both defined, but $AB\ne BA$ since $AB$ and $BA$ don't even have the same size.
- If $A=\mat{1&2\\3&4\\5&6}$, $B=\mat{2&1&1\\1&2&0\\1&0&2\\1&0&2\\2&2&1}$ and $C=\mat{1&3&0&7\\0&4&6&8}$, then $A$ is $3\times 2$, $B$ is $4\times 3$ and $C$ is $2\times 4$, so
- $AB$, $CA$ and $BC$ don't exist (i.e., they are undefined);
- $AC$ exists and is $3\times 4$;
- $BA$ exists and is $4\times 2$; and
- $CB$ exists and is $2\times 2$.
- In particular, $AB\ne BA$ and $AC\ne CA$ and $BC\ne CB$, since in each case one of the matrices doesn't exist.
- If $A=\mat{0&1\\0&0}$ and $B=\mat{0&0\\1&0}$, then $AB=\mat{1&0\\0&0}$ and $BA=\mat{0&0\\0&1}$. So $AB$ and $BA$ are both defined and have the same size, but they are not equal matrices: $AB\ne BA$.
- If $A=0_{n\times n}$ is the $n\times n$ zero matrix and $B$ is any $n\times n$ matrix, then $AB=0_{n\times n}$ and $BA=0_{n\times n}$. So in this case, we do have $AB=BA$.
- If $A=\mat{1&2\\3&4}$ and $B=\mat{7&10\\15&22}$, then $AB=\mat{37&54\\81&118}=BA$, so $AB=BA$ for these particular matrices $A$ and $B$.
- If $A=\mat{1&2\\3&4}$ and $B=\mat{6&10\\15&22}$, then $AB=\mat{36&54\\78&118}$ and $BA= \mat{36&52\\81&118}$, so $AB\ne BA$.
Commuting matrices
We say that matrices $A$ and $B$ commute if $AB=BA$.
Which matrices commute? Suppose $A$ is an $n\times m$ matrix and $B$ is an $\ell\times k$ matrix, and $A$ adn $B$ commute, i.e., $AB=BA$.
- $AB$ must be defined, so $m=\ell$
- $BA$ must be defined, so $k=n$
- $AB$ is an $n\times k$ matrix and $BA$ is an $\ell\times n$ matrix. Since $AB$ has the same size as $BA$, we must have $n=\ell$ and $k=m$.
Putting this together: we see that if $A$ and $B$ commute, then $A$ and $B$ must both be $n\times n$ matrices for some number $n$. In other words, they must be square matrices of the same size.
Examples 4 and 5 above show that for some square matrices $A$ and $B$ of the same size, it is true that $A$ and $B$ commute. On the other hand, examples 3 and 6 show that it's not true that square matrices of the same size must always commute.