$$X$$ and $$Y$$ are non-zero square matrices of size $$n \times n$$. If $$XY = {O_{n \times n}}$$ then

A real $$n \times n$$ matrix $$A$$ $$ = \left[ {{a_{ij}}} \right]$$ is defined as
follows $$\left\{ {\matrix{ {{a_{ij}} = i,} & {\forall i = j} \cr { = 0,} & {otherwise} \cr } .} \right.$$

The sum of all $$n$$ eigen values of $$A$$ is

Consider the differential equation $${{dy} \over {dx}} + y = {e^x}$$ with $$y(0)=1.$$ Then the value of $$y(1)$$ is

The integral $$\int\limits_{ - \alpha }^\alpha \delta \left( {t - {\pi \over 6}} \right)6\,\sin \,\left( t \right)dt$$ evaluates to