TITLE: Are inverse matrices unique?
QUESTION [14 upvotes]: Does a matrix have only one inverse matrix (like the inverse of an element in a field)? If so, does this mean that
$A,B \text{ have the same inverse matrix} \iff A=B$?

REPLY [7 votes]: If $A$, $B$ are square matrices with same inverse $C$, then $AC=CA=I$ and $BC=CB=I$. Therefore,
$$
        A =AI= A(CB)= (AC)B = IB = B.
$$
The odd thing about matrices is this: If $A$, $B$ are $n\times n$ matrices over a field, then $AB=I$ iff $BA=I$. This is a direct consequence of the fact that the $N\times N$ matrices form a finite-dimensional linear space.