A depletion type N -channel MOSFET is biased in its linear region for use as a voltage controlled resistor. Assume threshold voltage V_TH = -0.5 V, V_GS = 2.0 V, V_DS = 5 V, W/L=100, C_OX=10^-8 F/cm² and $${\mu _n}$$ = 800 cm²/V-s. The value of the resistance of the voltage controlled resistor (in $$\Omega $$ ) is _____.

For matrices of same dimension $$M,N$$ and scalar $$c,$$ which one of these properties DOES NOT ALWAYS hold ?

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 _______.

$$A$$ 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 _______.