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. 

Another way to look at the problem.
Consider the geometric multiplicity of the matrix. It has two linearly independent eigenvectors corresponding to zero and so the geometric multiplicity is equal to its algebraic multiplicity. Therefore the matrix is diagonalizable. But the diagonal form is the zero matrix and any vector similar to the zero matrix is still just the zero matrix.
