If B is an idempotent matrix, and A = I - B, then

B is an idempotent matrix∴ B2=BNow, A2=(I−B)2            =(I−B)(I−B)=I−IB−IB+B2=I−B−B+B2=I−2B+B2=I−2B+B=I−B=ATherefore, A is idempotent. Again,AB=(I−B)B=IB−B2=B−B2=B2−B2=OSimilarly, BA=B(I−B)=BI−B2=B−B=O.