Basis for Matrix A 
Find a basis for all $2\times2$ matrices $A$ for which
  $A\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ = $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$.

Maybe I'm dumb-- but isn't $A$ just the $0$ matrix? In which case, the base is simply the $0$ matrix as well?

Your guess is very intuitive, but let's check this rigorously:
$$\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}1&1\\1&1\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$$
and we end up with a system of equations $$\begin{cases}a+b=0\\a+b=0\\c+d=0\\c+d=0\end{cases}$$
Do you think you can find a basis for the solution space?
