Q:

How to get a zero matrix from matrix multiplication.

Let $A = \left[ \begin{matrix} 3 & 2 & 1\\ 5 & 0  &1\end{matrix}\right]$, 
how can I know if there is a matrix $N$ , st. $AN=0$ (N is not a zero matrix) 

A:

Try to solve 
$$A\pmatrix{x_1\\x_2\\x_3} = \pmatrix{0\\0}$$

in the question they demand $N$ be at least two columns.

And it doesn't matter how many columns $N$ should have. Say, from the above equation, you found a 3-element column vector $\mathbf x \ne \pmatrix{0&0&0}^T$ such that
$$A\mathbf x = \pmatrix{0\\0}$$
Then there would exist a $3\times j$ matrix $N = \mathbf x\pmatrix{c_1&c_2&\cdots&c_j}$, $c_1, \ldots, c_j\in\mathbb R$, such that
$$\begin{align*}
AN &= A\mathbf x\pmatrix{c_1&c_2&\cdots&c_j}\\
&= \pmatrix{0\\0}\pmatrix{c_1&c_2&\cdots&c_j}\\
&= \pmatrix{0&0&\cdots&0\\0&0&\cdots&0}
\end{align*}$$

Okay, another question. Is it possible to find matrix $N$ such that $AN$ and $NA$ are zero matrices of their respective dimensions, without $N$ being the zero matrix?

Similarly, try to solve
$$\begin{align*}
\pmatrix{y_1&y_2} A &= \pmatrix{0&0&0}\\
A^T \pmatrix{y_1\\y_2} &= \pmatrix{0\\0\\0}
\end{align*}$$
If there were one solution of non-zero row vector $\mathbf y^T$ that satisfies $\mathbf y^T A = \pmatrix{0&0&0}$, then can you construct an $N$ from $\mathbf y^T$ and $\mathbf x$ that satisfies your condition? And if you were not able to find such non-zero $\mathbf y^T$, can you reason that no non-zero $N$ would satisfy $NA = 0$?
If you notice, trying to find non-trivial solution for $\pmatrix{y_1&y_2}A = \pmatrix{0&0&0}$ can be considered as trying to determine whether the rows of $A$ are dependent. Can you see from $A$ if that is the case?

