If $$A = \left[ {\matrix{ 1 & { - 2} & { - 1} \cr 2 & 3 & 1 \cr 0 & 5 & { - 2} \cr } } \right]$$ and $$adj (A)$$ $$ = \left[ {\matrix{ { - 11} & { - 9} & 1 \cr 4 & { - 2} & { - 3} \cr {10} & k & 7 \cr } } \right]$$ then $$k=$$

Find the eigen values and eigen vectors of the matrix $$\left[ {\matrix{ 3 & { - 1} \cr { - 1} & 3 \cr } } \right]$$

$$A = \left[ {\matrix{ 2 & 0 & 0 & { - 1} \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 3 & 0 \cr { - 1} & 0 & 0 & 4 \cr } } \right].$$ The sum of the eigen values of the matrix $$A$$ is

A set of linear equations is represented by the matrix equations $$Ax=b.$$ The necessary condition for the existence of a solution for this system is