Exercise Prove that if row operations change [A B] to [I X], then X = A-1B.
Answer Since the same row operations change A to I, by this criterion, we see that A is invertible.
Let the i-th columns of B and X be bi and xi. Then the same operations change [A bi] to [I xi]. Equivalently, Ax = bi and Ix = xi have the same solutions. This implies Axi = bi. Then by this formula, AX = [Ax1 Ax2 ... Axn] = [b1 b2 ... bn] = B. Since A is invertible, we get X = A-1B.