True or False: Let $n ≥ 4$. There is an $n\times n$ matrix A such that $A^4 = 0$ and $\text{rank}(A+I_n) < 4$.

Question

I don't know how to judge this and I am a beginner at linear algebra so I need an answer which explains all necessary terms.

Answer 1

If $\text{rank}(A + I_n) < 4$, then by the rank-nullity theorem it's nullity is at least $1$, i.e. there exists some non-trivial vector $v \neq 0$ such that $(A + I_n) v = 0 \implies A v = -v$. In other words, $v$ would be an eigenvector of $A$ with eigenvalue $-1$. But then $$A^4 v = A(A(A(Av))) = v \neq 0$$ But why would this lead to a contradiction? (Hint: We haven't yet used the fact that $A^4 = 0$.)