Determine \(a\) so that the following system has a nontrivial solution:

\[
\left[\begin{array}{cccc}
2 & -1 & 0 & 0 \\
1 & a & 0 & 0 \\
0 & 0 & 2 a & 1 \\
0 & 0 & -1 & \frac{1}{2}
\end{array}\right]\left\{\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4}
\end{array}\right\}=\left\{\begin{array}{l}
0 \\
0 \\
0 \\
0 \end{array}\right\}
\]