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

Question

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

Vinay Kumar Sharma · Accepted Answer

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