For the function $$f\left( x \right) = {x^2}{e^{ - x}},$$ the maximum occurs when $$x$$ is equal to

If $$S = \int\limits_1^\infty {{x^{ - 3}}dx} $$ then $$S$$ has the value

If $$R = \left[ {\matrix{ 1 & 0 & { - 1} \cr 2 & 1 & { - 1} \cr 2 & 3 & 2 \cr } } \right]$$ then the top row of $${R^{ - 1}}$$ is

In the matrix equation $$PX=Q$$ which of the following is a necessary condition for the existence of atleast one solution for the unknown vector $$X.$$