Matrix
\(\begin{bmatrix} w & x \\ y & z \end{bmatrix}\) \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\) \(= \begin{bmatrix} \cos A \\ \sin A \end{bmatrix}\)
\(\begin{bmatrix} w & x \\ y & z \end{bmatrix}\) \(\begin{bmatrix} 0 \\ 1 \end{bmatrix}\) \(= \begin{bmatrix} -\sin A \\ \cos A \end{bmatrix}\)
\(\Rightarrow w = \cos A,\) \(y = \sin A,\) \(x = -\sin A,\) and \(z = \cos A\)
\(M = \begin{bmatrix} \cos A & \sin A \\ -\sin A & \cos A \end{bmatrix}\)
Eigenvalues
\(\det (\lambda I - A) = \begin{vmatrix} (\lambda - \cos A) & -\sin A \\ \sin A & (\lambda - \cos A) \end{vmatrix}\)
\(= (\lambda - \cos A)^2 + \sin^2 A\)
\(= \lambda^2 -2 (\cos A) \lambda + \cos^2 A + \sin^2 A\)
\(= \lambda^2 -2 (\cos A) \lambda + 1\)
\(\det (\lambda I - A) = 0\)
\(\rightarrow \lambda^2 -2 (\cos A) \lambda + 1 = 0\)
\(\rightarrow \lambda = \frac{1}{2}\left(2 \cos A \pm \sqrt{4 \cos^2 A - 4}\right)\)
\(\rightarrow \lambda = \cos A \pm \sqrt{\cos^2 A - 1}\)
\(\rightarrow \lambda = \cos A \pm \sqrt{-(1 - \cos^2 A)}\)
\(\rightarrow \lambda = \cos A \pm \sqrt{-\sin^2 A}\)
\(\rightarrow \lambda = \cos A \pm (\sqrt{-1}) \sin A\)
\(\rightarrow \lambda = \cos A \pm i \sin A\)