\begin{pmatrix} 2na & -a & -a & -a & -a & -a& -a\\ -a& a+b & 0 & 0 & -b & 0 & 0\\ -a& 0 & a+b & 0 & 0 & -b &0 \\ -a& 0 & 0 & a+b & 0 & 0&-b \\ -a& -b & 0 & 0 & a+b & 0 & 0\\ -a& 0&-b & 0 & 0 & a+b &0 \\ -a& 0& 0&-b & 0 & 0 & a+b \end{pmatrix}
Above matrix is for $n=3$. There are block matrices $(a+b)I_3$ and $(-b)I_3$ above. I think the characteristic polynomial of this matrix for any natural number $n$ is calculated efficiently.
The rank is $2n$, so the characteristic polynomial should zero as the constant part. But I cannot calculate the determinant and I can't get zero when I put $t=0$ when I calculate $\text{char}(M)(t)$ for any $n$.
Also, I would like to know about the form, for example, spectral decomposition or Jordan from to compute eigenvalue of this matrix rapidly. Please give me a favor.