A $n \times n$ matrix if $A^2x=0$ has non trivial solution.

Question

Then show that $Ax=0$ also has non trivial solution. Is there a solution considering idea of rank and rref.

Answer 1

A linear map $T:X \to Y$ is injective if and only if its kernel is trivial. Hence, if $\ker (A^2) \neq \{0\}$, then $A^2$ is not injective. If $A$ were injective (its kernel would be trivial), then $A^2=A \circ A$ would be injective as well, a contradiction.