Quantum mechanics characteristic polynomial

Question

So I am having a issue with this question and not sure if I am doing something wrong, or the matrix is wrong.

I am going to attach a picture of the question as I want to make sure the question is conveyed correctly

It is only part a) my working are as follows:

Starting with the Hamiltonian

$$\hat{H}\:\:E_n\mid E\,\rangle= E_n\mid E\,\rangle$$

Re-arranging and applying the identity matrix $$\hat{H}\:\:E_n\mid E\,\rangle-\:E_n\mid E\,\rangle=0$$

simplifying $$\hat H -IE_n=0$$

$$\begin{pmatrix}\mu &-\mu \\ -\mu &2\mu \end{pmatrix}-E_n\begin{pmatrix}1&0\\ 0&1\end{pmatrix}=0$$

subtracting the matrices and finding the detriment I have the following characteristic equation.

$$\left(\mu -E_n\right)\left(2\mu -E_n\right)-\left(-\mu \right)\left(-\mu \right)=E_{n}^2-3\mu E_n+\mu ^2=0$$

So this is where I am stumped I don't know if I have been looking at the problem too long, but I can see to find any roots that are associated with this equation

$$E_{n}^2-3\mu E_n+\mu ^2=0$$

Answer 1

Hint: apply the quadratic formula. $a = 1$, $b = -3\mu$, and $c = \mu^2$. $\mu$ is in fact, just a number. Your two energy values will just pop out as a function of $\mu$.

Answer 2

As the Hint in your questions states, the energies are the eigenvalues of the Hamiltonian $\hat{H}$. $E_0$ is the eigenvalues with the lowest energy, and it must be a function of $\mu$, i.e. $E_0(\mu)$.