1. The problem statement, all variables and given/known data Suppose A and I are n*n matrices and I is a unit matrix ,and A is an idempotent matrix,ie, A=A^2 . Show that if rankA=r and rank(A-I)=s,then r+s=n 2. Relevant equations no 3. The attempt at a solution I know that if A is an idempotent matrix ,it will have eigenvalues either 0 or 1. (Proof: Ax=(A^2)x ,and Ax=λx so(A^2)x = A(Ax)=Aλx=λ(Ax)=(λ^2) x thus, λx=(λ^2) x →(λ-1)λx=0. Suppose x is a nonzero eigenvector, λ = 1 or 0. ) that is, if x1 and x2 are eigenvectors associated with eigenvalue 0 and 1 respectively, then, A(x1)=0(x1)=0, (a) and A(x2)=1(x2)=x2 so(A-I)(x2)=0 (b) Now, I have (a) and (b) , how to show that rankA+rank(A-I)=n?