Theorem: If A is a nilpotent square matrix (that is for some natural number k>0, A^k =0) then (I + A) is an invertible matrix. Pf: Let B denote the inverse which will constructed directly. Let n be the smallest integer so that 2^n>k. B=(1-A)(1+A^(2^1))(1+A^(2^2))....(1+A^(2^n)) then, B(1+A)=(1+A)B=1-A^(2^n)=1 - (A^(k))(A^(2n-k)) = 1 now if QED I wonder if this is a sufficient proof? I've never taken a linear algebra course so I really don't know!