Question

If A and B are two square matrices such that B=−A−1BA, then (A+B)2 is equal to

Answer 1

B=-A-BA
B= -A(1+B)
-B/A=1+B
B=-B/A-1
B=(A+B)/-A
(A+B) = -AB
Multiplying 2 both sides
2(A+B) = -2AB

Answer 2

It  is given that    

                               B = -A^-1(BA)

          now, pre-multiplying by A on both sides, we will get

                                AB = -BA

                               AB+BA = 0.........................(1)

Now coming to the problem:

(A+B)2=(A+B)*(A+B)  =  A^2+B^2+AB+BA.....................(2)

from (1) and (2), (SUBSTITUTING AB= -BA)  we get

( A+B)2=A^2+B^2