<s>[INST] 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. 
 [/INST] Note that solving $XA=0$ is equivalent to solving $A^\top Y=0$. Since 
$$
\DeclareMathOperator{rref}{rref}\rref A^\top = 
\left[\begin{array}{rrrr}
1 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
it follows that all solutions to $A^\top \vec y=\vec 0$ are of the form
$$
\vec y=\lambda
\left[\begin{array}{r}
1 \\
1 \\
-1 \\
0
\end{array}\right]
$$
Now, let $\lambda_1,\dotsc,\lambda_4$ be your four favorite numbers. Then
$$
Y=
\begin{bmatrix}
\lambda_1\vec b & \lambda_2\vec b & \lambda_3\vec b & \lambda_4\vec b 
\end{bmatrix}
$$
is a $4\times 4$ matrix that solves $A^\top Y=0$ where 
$$
\vec b=
\left[\begin{array}{r}
1 \\
1 \\
-1 \\
0
\end{array}\right]
$$
Taking $X=Y^\top$ then solves your original problem.
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