Information about $T$ if $T^2=0$ What can we say about $T$ if $T^2=0$? How can we prove that it has zero as an eigenvalue?

If $T^2=0$, then $det(T^2)=det(0)=0$.  But as $det(AB)=det(A)det(B)$, we now see that $det(T)^2=0$, so $det(T)=0$.  As $det(T)$ is the product of all eigenvalues for $T$, we see that $0$ is an eigenvalue of $T$. 
