if CA= I, and AB = I, then B = C:
C =C I = C(AB) = (CA)B = IB = B.
thus B ( = C) is "one inverse" for A. suppose B' is another, so that B'A = AB' = I:
B' = B'I = B'(AB) = (B'A)B = IB = B.
so a matrix that is invertible can only have one inverse, and if a matrix has a left-inverse and a right-inverse, both the left-inverse and the right-inverse
are the same matrix, A^-1.
so to find an inverse for A, you can just find a right-inverse or a left-inverse, which saves time checking by matrix multiplication.
personally, i would check det(A) for this particular problem, because A^-1 exists iff det(A) ≠ 0.