Let $A = \pmatrix {a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}}$
Let $A \neq 0$ then construct a nilpotent matrix such that (1) $A^2 = 0$
(2) $A^2 \neq 0$ and $A^3 = 0$
Also find linearly independent eigen vectors in each case.
I have tried by multiplying matrix $A$ to itself and equated each element of $A^2$ to $0$, it becomes very tedious. Please help me.