examination problem

Question

let N be a non-zero 3 x 3 matrix with property $N^2 = 0 $ , then show that N has one non-zero eigen vector ., ,..,.,.,.,.,., accumulating all the the knowledge that i have about linear algebra , the question still seems too general for me to reach relevant approach .

Answer 1

If $\lambda$ is an eigenvalue of $N$, then $\lambda^2$ is an eigenvalue of $N^2$, hence $\lambda^2=0$ and therfore $\lambda=0$.

Thus, the only possible eigenvalue of $N$ is $0$.

Since $N \ne 0$, there is $x \ne 0$ with $y:=Nx \ne 0$.

Result: $Ny=N^2x=0$