TITLE: Prove that $A+I$ is invertible if $A$ is nilpotent
QUESTION [17 upvotes]: Possible Duplicate:
Units and Nilpotents 

Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).

REPLY [4 votes]: A matrix $A$ is nilpotent if and only if all its eigenvalues are zero. It is not hard also to see that the eigenvalues of $A+I$ will all be equal to $1$ (when we add $I$ to any matrix, we just shift its spectrum by 1). Thus $A+I$ is invertible,
since all its eigenvalues are non-zero.