Hey everyone i have no clue how to solve this one, please help.
If B is any nilpotent matrix prove that I-B is invertible and find a formula for (I-B)^-1 in terms of powers of B?
If B is nilpotent then B^N=0 for some positive integer N. Thus the series $\displaystyle I+B+B^2+B^3+\ldots$ finishes with the term B^{N–1}, since all the subsequent terms are zero. That suggests that $\displaystyle (I-B)^{-1} = I+B+B^2+\ldots+B^{n-1}$, a conjecture which you can verify by checking that $\displaystyle (I-B)(I+B+B^2+\ldots+B^{n-1}) = (I+B+B^2+\ldots+B^{n-1})(I-B) = I$.