I believe I might be having a bit of trouble with a question that asks me to find the values of $a$ and $b$ such that the polynomial $x^2-bx-a$ is irreducible and the matrix $\begin{bmatrix} 0 & a \\ 1 & b \end{bmatrix}$ has order $8$.
This is not a homework question in any class that I am taking. This is regarding a research project that I am planning on working with a professor for which I have almost no clue what I am doing.