-1
$\begingroup$

On several occasions, I have heard that matrix multiplication is commutative for square matrices $A$ and $B$ when they represent linear transformations. Is this true? I know that in general $AB$ is not $BA$ for some matrices $A$ and $B$.

$\endgroup$
  • $\begingroup$ It's not true even if the matrices are $1\times1$, if the scalar ring is not commutative. Over a commutative ring, I think you know the answer well: matrix multiplication of square matrices of size $n$ is not commutative, unless $n=1$. The classical example is $A=\pmatrix{0&1\\ 0&0}$ and $B=\pmatrix{1&0\\ 0&0}$. This is all about the rules of matrix multiplication. It doesn't matter if the matrices are representations of some linear transformations or not. $\endgroup$ – user1551 Feb 12 '15 at 9:08
4
$\begingroup$

All $n \times n$ square matrices (say, over the field $\mathbb{F}$) can be regarded as linear transformations of an $n$-dimensional vector space $\mathbb{V}$ over $\mathbb{F}$ in a given basis, which allows us to identify $\mathbb{V} \cong \mathbb{F}^n$. In particular, as you note in general $AB$ and $BA$ differ, and hence the answer is no.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.