TITLE: If $(A + I)^4 = 0$, $A$ is non-singular
QUESTION [8 upvotes]: If $A$ be an n × n real matrix such that $(A + I)^4 = 0$, where $I$ denotes the identity matrix, how to prove that $A$ is non-singular?

REPLY [9 votes]: If $A$ were singular, then there would be a non-zero vector $x$ with $Ax=0$ and therefore $(A+I)x=0+x=x$.  But then $(A+I)^4x=x$, whereas we're given that $(A+I)^4x=0$.  (This is essentially Jlamprong's answer, without using the word "eigenvalue" or the letter $\lambda$.)