detA·detB = det(AB) = det(E) = 1
所以det(A) ≠ 0
所以A可逆
A·B = E
设B'·A = E
则B' = B'·E = B'·(A·B) = (B'·A)·B = E·B = B
所以 AB = BA = E
所以A的逆矩阵等于B