If the matrix $A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1 \end{pmatrix}$ satisfies the equation $A^{20} + \alpha\,A^{19} + \beta\,A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ for some real numbers $\alpha$ and eta, then $\beta - \alpha$ is equal to: