Because A+B=En, (A+B) * (A-B) = A 2-B 2 = A-B, transpose A 2-A = B 2-B, and substitute B=E-A into the right end of the above formula, and A 2-A = (E-). This proof is more complicated, but you can find this formula in general books, in the online textbook of Tongji University Press. Because r(A)+r(B)=n, then r(AB)=0, so AB=BA=0, and A*(E-A)=0, so A*A=A, and likewise B * B =