Why $\textbf{AB=BA}$ when $\textbf{B=A^{-1}}$? [duplicate]
Matrix multiplication is not commutative:
$$AB \neq BA$$
So how come that when B is inverse of A $B=A^{-1}$ suddenly the commutative law works since:
$$AB=I_n=BA$$
when $B=A^{-1}$?
I understand there can be exceptions to all rules but why is this case an exception? What is the intuition behind this result?
Solution 1:
The definition of an inverse of $A$ is a matrix $B$ such that
$$AB=BA=I.$$
Such a matrix $B$ of course need not exist, but when it does, $A$ has this special property that we call being "invertible." End of story.