Let $$A = \left[ {{a_{ij}}} \right],\,\,1 \le i,j \le n$$ with $$n \ge 3$$ and
$${{a_{ij}} = i.j.}$$ Then the rank of $$A$$ is

Identity which one of the following is an eigen vectors of the matrix $$A = \left[ {\matrix{ 1 & 0 \cr { - 1} & { - 2} \cr } } \right]$$

Let $$A$$ be $$3 \times 3$$ matrix with rank $$2.$$ Then $$AX=O$$ has

The necessary condition to diagonalize a matrix is that