Matrix Problem (Proving Invertibility)

In this exercise A is a square matrix.

1) Show that if A^2 = 0 then (A - I) is invertible.

2) Show that if A^3 = 0 then (A - I) is invertible.

3) Show that if A^4 = 0 then (A - I) is invertible.

4) Assume that there is a natural number k such that A^k = 0. Prove that (A - I)

Expanding on previous help

Formula that might be of use: remember the geometric series formula.

$\displaystyle \frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$ for |x|<1. But this is just to ensure your series converges. In this case your matrix is nilpotent, so your infinite series in fact is not quite so infinite.