let A be an nxn matrix such that A^2=0nxn.Prove that if y is an eigenvalue of A,then we must have that y=0
Since $\displaystyle y$ is eigenvalue of $\displaystyle A$ it means there is $\displaystyle \bold{0}\not =\bold{x}\in \mathbb{R}^n$ so that $\displaystyle A\bold{x} = y\bold{x}$.
Thus, $\displaystyle A(A\bold{x}) = A(y\bold{x}) \implies A^2 \bold{x} = y(A\bold{x})\implies \bold{0} = y(y\bold{x}) \implies y^2\bold{x} = 0$
From here we see that $\displaystyle y=0$ because $\displaystyle \bold{x}\not = \bold{0}$.