If A and B are two square matrices of the same order such that AB = A and BA = B , then (A + B)^2 is equal to:

Step-by-step solution

Given AB = A and BA = B . We can find A^2 and B^2 as follows: A^2 = A A = (AB)A = A(BA) Substituting BA = B , we get: A^2 = AB = A Similarly, for B^2 : B^2 = B B = (BA)B = B(AB) Substituting AB = A , we get: B^2 = BA = B Now, expanding (A + B)^2 : (A + B)^2 = (A + B)(A + B) (A + B)^2 = A^2 + AB + BA + B^2 Substituting A^2 = A , B^2 = B , AB = A , and BA = B into the expansion: (A + B)^2 = A + A + B + B (A + B)^2 = 2A + 2B = 2(A + B) Answer: 2(A + B)