The maximum value of $$'a'$$ such that the matrix $$\left[ {\matrix{ { - 3} & 0 & { - 2} \cr 1 & { - 1} & 0 \cr 0 & a & { - 2} \cr } } \right]$$ has three linearly independent real eigenvectors is

A system matrix is given as follows $$$A = \left[ {\matrix{ 0 & 1 & { - 1} \cr { - 6} & { - 11} & 6 \cr { - 6} & { - 11} & 5 \cr } } \right].$$$

The absolute value of the ratio of the maximum eigenvalue to the minimum eigenvalue is ___________.

$$A = \left[ {\matrix{ p & q \cr r & s \cr } } \right];B = \left[ {\matrix{ {{p^2} + {q^2}} & {pr + qs} \cr {pr + qs} & {{r^2} + {s^2}} \cr } } \right]$$
If the rank of matrix $$A$$ is $$N$$, then the rank of matrix $$B$$ is

The equation $$\left[ {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr } } \right]$$ has