Defining the $2 \times 2$ transfer matrix
\begin{align}\tag{1}
Q = \begin{pmatrix} -\lambda & 1 \\ -1 & 0 \end{pmatrix},
\end{align}
the characteristic polynomial (CP) of the $M \times M$ matrix $A_M$ is given by
\begin{align}
P_M(\lambda) &= \langle 1{-}\lambda, 2| \, Q^M \,|1,0\rangle \tag{2a}\\
&= 2 T_M(\lambda/2) - U_{M-1}(\lambda/2) \tag{2b}\\
&= 2\cos(M \varphi) + \frac{\sin(M \varphi)}{\sin(\varphi)}\tag{2c},
\end{align}
with Chebyshev polynomials $T_M,U_M$, and with $\lambda=-2\cos\varphi$.

The unnormalized right eigenvectors $A_M x_\mu=\lambda_\mu x_\mu$ fulfill
\begin{align}
x_{\mu,m} = (-1)^m\langle 1, 0| \, Q^m \, |1,0\rangle.\tag{3}
\end{align}
The eigenvector normalization can be related to the derivative $P_M'(\lambda)$, see, e.g., https://arxiv.org/abs/2103.10776 for details.

Regarding to your question, I don't think that a closed form expression exist for $\lambda_\mu$ if $M>5$, as due to the left boundary term, the CP does not factorize in this case. However, it is often not necessary to explicitly calculate the eigenvalues, see my cited paper, as the CP together with the eigenvectors contains enough, or even more, information.