if AB r 2 matrices AB=B BA=A then A^2+B^2=

We are given that A and B are two matrices such that AB=B and BA=A.

so A^2 + B^2=?

A^2=AA , B^2=BB

A^2 + B^2 = AA+ BB

       =(BA)(BA) + (AB)(AB)       [since BA=A and AB=B]

        =B(B)A + A(A)B            [since BA=A and AB=B]

        =B(A) + A(B)

        =A + B                  [since BA=A and AB=B]


Hence    A^2 +B^2  =  A+B

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