The rank of the matrix $$M = \left[ {\matrix{ 5 & {10} & {10} \cr 1 & 0 & 2 \cr 3 & 6 & 6 \cr } } \right]$$ is

Consider the $$5 \times 5$$ matrix $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 & 5 \cr 5 & 1 & 2 & 3 & 4 \cr 4 & 5 & 1 & 2 & 3 \cr 3 & 4 & 5 & 1 & 2 \cr 2 & 3 & 4 & 5 & 1 \cr } } \right]$$
It is given that $$A$$ has only one real eigen value. Then the real eigen value of $$A$$ is

The value of $$x$$ for which the matrix $$A = \left[ {\matrix{ 3 & 2 & 4 \cr 9 & 7 & {13} \cr { - 6} & { - 4} & { - 9 + x} \cr } } \right]$$ has zero as an eigen value is __________.

Let $${M^4} = {\rm I}$$ (where $${\rm I}$$ denotes the identity matrix) and $$M \ne {\rm I},\,\,{M^2} \ne {\rm I}$$ and $${M^3} \ne {\rm I}$$. Then, for any natural number $$k, $$ $${M^{ - 1}}$$ equals: