Answer
\begin{pmatrix} \frac{3}{2}\\ 1 \end{pmatrix}
Work Step by Step
The problem states to find one nontrivial solution by inspection.
\begin{equation}
A = \begin{pmatrix} 4 & -6 \\ 8 & -12 \\ 6 & -9 \end{pmatrix} = \begin{pmatrix} 0\\ 0\\0 \end{pmatrix}
\end{equation}
We notice that the second column is the first column multiplied by $-\frac{3}{2}$. Thus, one possible solution is multiplying the first column by $\frac{3}{2}$ and the second column by 1, and adding them.
Thus, the solution, in vector form, is
\begin{pmatrix} \frac{3}{2}\\ 1 \end{pmatrix}
