You can use determinants to prove it. You know if det(A) = 0, an inverse does NOT exist. So if you take det() of each side:

det(2AB - 3AC) = det(I)

det(A)*(det(2B - 3C)) = 1

Can only be true if det(A) is nonzero, so the inverse must exist.

Oh, and to get the second part, multiply both sides by A inverse.