Proof: Nilpotent Matrix A & Real Matrices AB-BA=I

[kumarrukhaiyar]

1. A square matrix A is called nilpotent if a^k = 0 for some k > 0. Prove that if A is nilpotent, then I + A is invertible.

2. Show that the equation AB - BA = I has no solutions in n x n matrices with real entries.

 

[radou]

Regarding 2, try to use the matrix trace function.

 

[matt grime]

1) Forget matrices. You want to find

(1+x)^{-1}

well, what is that as a power series?