Find a nonzero matrix, so that when it is multiplied by another nonzero matrix, the zero matrix is the result. Here's an interesting review question I have: Find a nonzero matrix, so that when it is multiplied by another nonzero matrix, the zero matrix is the result. How would you go about solving this? Thanks!
 A: First, I’d look for a simple example, so I’d start with $2\times 2$ matrices, say $A$ and $B$. If I make the bottom row of $A$ all zeroes, the bottom row of $AB$ will automatically be all zeroes. If I make the second column of $B$ all zeroes, the second column of $AB$ will automatically be all zeroes. At this point I have
$$\pmatrix{a&b\\0&0}\pmatrix{c&0\\d&0}=\pmatrix{ac+bd&0\\0&0}\;,$$
and all that’s needed in order to finish the job is to find $a,b,c$, and $d$ so that $ac+bd=0$, at least one of $a$ and $b$ is non-zero, and at least one of $c$ and $d$ is non-zero; this is very easy to do.
If you’re comfortable thinking of matrices as linear transformations, you can look for a pair of non-zero linear transformations $S$ and $T$ such that $T\circ S$ sends everything to $0$. Here again, I’d try to keep it simple, so I’d look for linear transformations sending $R^2$ to itself. What if $S$ is the linear transformation that projects the plane to the $x$-axis, and $T$ is the one that projects the plane to the $y$-axis? Clearly, $T\circ S$ (and for that matter $S\circ T$) sends everything to the origin, so it’s the zero transformation, but neither $S$ nor $T$ by itself does so. Having seen that, all I’d have to do is write down the matrices of $S$ and $T$.
A: You can find surprising results with matrices of nullity 2 or higher. For example, here are two matrices that multiply to zero, despite having all non-zero entries: 
$\begin{bmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12 \\
13 & 14 & 15 & 16 \\
\end{bmatrix}
*
\begin{bmatrix}
4 & 3 & -2 & -9 \\
-5 & -5 & 1 & 14 \\
-2 & 1 & 4 & -1 \\
3 & 1 & -3 & -4 \\
\end{bmatrix}
=
\begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{bmatrix}
$
