For $$A = \left[ {\matrix{ 1 & {\tan x} \cr { - \tan x} & 1 \cr } } \right],$$ the determinant of $${A^T}\,{A^{ - 1}}$$ is

The value of $$'x'$$ for which all the eigenvalues of the matrix given below are real is $$\left[ {\matrix{ {10} & {5 + j} & 4 \cr x & {20} & 2 \cr 4 & 2 & { - 10} \cr } } \right]$$

Consider system of linear equations : $$$x-2y+3z=-1$$$ $$$x-3y+4z=1$$$ and $$$-2x+4y-6z=k,$$$

The value of $$'k'$$ for which the system has infinitely many solutions is _______.

The value of $$'P'$$ such that the vector $$\left[ {\matrix{ 1 \cr 2 \cr 3 \cr } } \right]$$ is an eigenvector of the matrix $$\left[ {\matrix{ 4 & 1 & 2 \cr P & 2 & 1 \cr {14} & { - 4} & {10} \cr } } \right]$$ is ________.