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. 

Solve the equation $A^Tx = 0$. If $A$ is not invertible then it must have a nonzero solution $\vec x_0$.
Then set $B = \begin{pmatrix} \vec x_0 \\ \vec x_0 \\ \dots \\ \vec x_0\end{pmatrix}$ -- the matrix having $\vec x_0$ as rows. Since $\vec x_0^T A = 0$, it means that $BA = 0$ and $B \neq 0$.
Since $\vec x_0$ is a vector with rational entries, we can represent it as $\vec x_0 = \begin{pmatrix} \dfrac{p_1}{q_1} &  \dfrac{p_2}{q_2} & \dots & \dfrac{p_n}{q_n}\end{pmatrix}$. Let $k = \operatorname{lcm}(q_1, q_2, \dots, q_n)$. Then $C = kB \neq 0$ is a desired matrix.
