If the matrix A = [1, 0, 0; 0, 2, 0; 3, 0, -1] satisfies the equation A^20+α A^19+β A = [1, 0, 0; 0, 4, 0; 0, 0, 1] for some real numbers α and β,…

If the matrix $\displaystyle \mathrm{A}=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1\end{array}\right]$ satisfies the equation $\displaystyle \mathrm{A}^{20}+\alpha \mathrm{A}^{19}+\beta \mathrm{A}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1\end{array}\right]$ for some real numbers $\displaystyle \alpha$ and $\displaystyle \beta$, then $\displaystyle \beta-\alpha$ is equal to \_\_\_\_.