Is it safe to say matrix multiplication is commutative ONLY if one matrix is the inverse of the other? The only way I could believe this was the 2 commutative type properties I know of:
$$AA^{-1}=A^{-1}A=I$$
and
$$adj A*A=A*adjA = |A|I$$
where $adj A$ is the adjoint of matrix A.
The first equation when expanded with $A^{-1}=\frac{adjA}{|A|}$ you end up getting the second equation. So both are the same equation. These are the only two properties that explains commutativity as far as I'm aware of(are there any more?) and my conclusion was based on these only. Also multiplication with identity matrix is also commutative but any other counter cases? So is it safe to say matrix multiplication is commutative ONLY if one matrix is the inverse of the other? Thanks!!
Best Answer
Not at all. For instance, the identity matrix commutes with every other square matrix of the same size. As another example, the matrices $\left[\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}0&0\\0&1\end{smallmatrix}\right]$ commute, but none of them is a multiple of the other one.