Find a non-zero integer matrix $X$ such that $XA=0$ where $X,A,0$ are all $4 \times 4$ Let $A$ be the following $4 \times 4$ matrix.
\begin{bmatrix}1&2&1&3\\1&3&2&4\\2&5&3&7\\1&4&1&5\end{bmatrix}
How can we find a non-zero integer matrix $C$ such that $CA = 0_{4 \times 4}$
Note that $0$ is a $4 \times 4$ matrix. 

One option is use the projector onto the $\color{red}{null}$ space 
$$
\color{red}{\mathbf{P}_{\mathcal{N}\left(\mathcal{A^{*}}\right)}} = \mathbf{I}_{4} - \mathbf{A}\mathbf{A}^{+}
$$
The target matrix and pseudoinverse are
$$
\mathbf{A} = 
\left(
\begin{array}{cccc}
 1 & 2 & 1 & 3 \\
 1 & 3 & 2 & 4 \\
 2 & 5 & 3 & 7 \\
 1 & 4 & 1 & 5 \\
\end{array}
\right),
\qquad
\mathbf{A}^{+}
=
\frac{1}{18}
\left(
\begin{array}{rrrr}
 26 & -19 & 7 & -9 \\
 -16 & 11 & -5 & 9 \\
 -12 & 15 & 3 & -9 \\
 10 & -8 & 2 & 0 \\
\end{array}
\right)
$$
The projector matrix is
$$
\color{red}{\mathbf{P}_{\mathcal{N}\left(\mathcal{A}\right)}} = \mathbf{I}_{4} - \mathbf{A}^{+}\mathbf{A} = 
\frac{1}{3}
\left(
\begin{array}{rrrr}
 1 & 1 & -1 & 0 \\
 1 & 1 & -1 & 0 \\
 -1 & -1 & 1 & 0 \\
 0 & 0 & 0 & 0 \\
\end{array}
\right)
$$
Verify:
$$
\color{red}{\mathbf{P}_{\mathcal{N}\left(\mathcal{A}\right)}} \mathbf{A}= 
\left(
\begin{array}{cccc}
 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 \\
\end{array}
\right)
$$
If you look carefully at the solution matrix, you will appreciate the insight of @gt6989b.
