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 ________.

$$A$$ real $$\left( {4\,\, \times \,\,4} \right)$$ matrix $$A$$ satisfies the equation $${A^2} = {\rm I},$$ where $${\rm I}$$ is the $$\left( {4\,\, \times \,\,4} \right)$$ identity matrix. The positive eigen value of $$A$$ is _______.

Consider the matrix $${J_6} = \left[ {\matrix{ 0 & 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1 & 0 & 0 \cr 0 & 0 & 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 & 0 & 0 \cr 1 & 0 & 0 & 0 & 0 & 0 \cr } } \right]$$

Which is obtained by reversing the order of the columns of the identity matrix $${{\rm I}_6}$$. Let $$P = {{\rm I}_6} + \alpha \,\,{J_6},$$ where $$\alpha $$ is a non $$-$$ negative real number. The value of $$\alpha $$ for which det $$(P)=0$$ is _______.