$$X$$ and $$Y$$ are non-zero square matrices of size $$n \times n$$. If $$XY = {O_{n \times n}}$$ then

The eigen values of a $$2 \times 2$$ matrix $$X$$ are $$-2$$ and $$-3$$. The eigen values of matrix $${\left( {X + 1} \right)^{ - 1}}\left( {X + 5{\rm I}} \right)$$ are

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

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