Consider a general \(2 \times 2\) matrix \(\mathbb{M}\) which we can express as

\[\mathbb{M}=\left(\begin{array}{ll}a & b \tag{2.99}\\c & d\end{array}\right)\]

for some, in general complex, numbers \(a,b, c, d\).

(a) What is the characteristic equation for this matrix; that is, what is the polynomial whose roots are eigenvalues of this matrix?

(b) What are the constraints on the element values \(a,b, c, d\) such that all eigenvalues are real numbers?

(c) Now, let's further impose the constraints that the determinant of \(\mathbb{M}\) is 1 and its trace (the sum of the elements on the diagonal) is 0 . What are the possible values for the eigenvalues now? If the eigenvalues are real, what is a possible form for the matrix (\mathbb{M}\) ?