# Math Help - Matrix Problem (Proving Invertibility)

1. ## 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)

2. When I first did this question, consider the inverse of (x-1) for a real number |x| <1 and see if you can construct an inverse

3. ## Expanding on previous help

Formula that might be of use: remember the geometric series formula.
$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$ 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.