Two linearly independent eigenvectors with eigenvalue zero What is the only $2\times 2$ matrix that only has eigenvalue zero but does have two linearly independent eigenvectors? 
I know there is only one such matrix, but I'm not sure how to find it. 

Let $A$ be any such matrix. Let $\beta=[\mathbf{v}_1,\mathbf{v}_2]$ be a basis made up of eigenvectors of $A$. If $P$ is the matrix that has $\beta$ in the columns, then $P^{-1}AP$ is diagonal, with the eigenvalues of $A$ in the diagonals. But such a matrix is
$$\left(\begin{array}{cc} 0&0\\0&0\end{array}\right).$$
So $PAP^{-1}=0$. Multiplying on the left by $P^{-1}$ and on the right by $P$, we get $A = P^{-1}0P = 0$. 
