The dimension of the null space of the matrix $$\left[ {\matrix{ 0 & 1 & 1 \cr 1 & { - 1} & 0 \cr { - 1} & 0 & { - 1} \cr } } \right]$$ is

Given that $$A = \left[ {\matrix{ { - 5} & { - 3} \cr 2 & 0 \cr } } \right]$$ and $${\rm I} = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right],$$ the value of $${A^3}$$ is

The matrix $$M = \left[ {\matrix{ { - 2} & 2 & { - 3} \cr 2 & 1 & 6 \cr { - 1} & { - 2} & 0 \cr } } \right]$$ has eigen values $$-3, -3, 5.$$ An eigen vector corresponding to the eigen value $$5$$ is $${\left[ {\matrix{ 1 & 2 & { - 1} \cr } } \right]^T}.$$ One of the eigen vector of the matrix $${M^3}$$ is

A real $$n \times n$$ matrix $$A$$ $$ = \left[ {{a_{ij}}} \right]$$ is defined as
follows $$\left\{ {\matrix{ {{a_{ij}} = i,} & {\forall i = j} \cr { = 0,} & {otherwise} \cr } .} \right.$$

The sum of all $$n$$ eigen values of $$A$$ is