Let $A$ be a real $3 \times 3 $ matrix such that rank$(A) = 2$.
Prove that $A^2 \neq 0_3$. where $0_3$ represents the null matrix of order $3$.
I am looking for a solution involving only basic manipulation using matrices. I already have a better solution using the range and the nullity of $A$.
Thank you in advance!
Edit. No Sylvester's inequality, Jordan form or range+nullity / linear transformations. At most use the definition of the rank as the dimension of the column/row space.