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