AB = A and BA = B, then (here A & B are matrix of n x n) which of the following must be true -
Step-by-Step Solution
Key Concept: Given AB = A and BA = B. We need to check if A^2 = A. Since A^2 = A(A) = A(BA) = (AB)A = AA = A^2. Wait, let's re-evaluate: A^2 = A(A) = A(BA) = (AB)A = AA = A^2. Actually, A^2 = A(AB) = (AB)B = AB = A. Thus A^2 = A.
Given AB = A and BA = B. Consider A^2 = A(A). Substituting A = AB, we get A^2 = A(AB) = (AA)B. This doesn't immediately show A^2 = A. Let's try A^2 = A(A) = A(BA) = (AB)A = AA = A^2. Actually, A^2 = A(A) = A(AB) = (AB)B = AB = A. Therefore, A^2 = A.
Correct Answer: B