4.3 Re-write the equations obtained in 4.2 to obtain a matrix equation, in the form \(M\vec{z}=\vec{0}\). Note that \(M\) is a 2-by-2 complex-valued matrix, and \(\vec{z}=\begin{pmatrix}z_1 \\ z_2\end{pmatrix}\) is a 2-component vector with complex-values. If \(M\) is (left-)invertible, meaning that we can find the (left-)inverse \(M^{-1}\), then we will obtain\(M^{-1}\left(M\vec{z}\right)=M^{-1}\left(\vec{0}\right)=\vec{0}\). But matrix multiplication is associative, meaning that \(M^{-1}\left(M\vec{z}\right)=\left(M^{-1}M\right)\vec{z}=\vec{z}\), and thus only the trivial solution \(z_1=0,\ z_2=0\) exists. Thus, we want to solve for the case where the determinant of \(M\) equals zero, so that we have non-trivial solutions.