Transmutation

Since we are analyzing the long-term behavior of a matrix, we should find the eigenvalues and eigenvectors of the matrix. Solving the characteristic equation of the matrix $(0.975 - \lambda)(0.6 - \lambda) - 0.4 * 0.025 = 0$ yields $\lambda_{1} = 1$ and $\lambda_{2} = 0.575$ . We can construct two sets of augmented matrices and use Gaussian elimination to find the eigenvectors associated with the above eigenvalues:

$\begin{array}{cc|c} 0.975 - \lambda_{1} & 0.4 & 0\\ 0.025 & 0.6 - \lambda_{1} & 0 \end{array} \to \begin{array}{cc|c} -0.025 & 0.4 & 0\\ 0.025 & -0.4 & 0 \end{array} \to \begin{array}{cc|c} 1 & -16 & 0\\ 0 & 0 & 0 \end{array}$

Solving this simple equation yields a possible eigenvector $\overrightarrow{v_{1}} = \begin{bmatrix} 16 \\ 1 \end{bmatrix}$ . Likewise, we can apply the same process for $\lambda_{2}$ to find an associated eigenvector:

$\begin{array}{cc|c} 0.975 - \lambda_{2} & 0.4 & 0\\ 0.025 & 0.6 - \lambda_{2} & 0 \end{array} \to \begin{array}{cc|c} 0.4 & 0.4 & 0\\ 0.025 & 0.25 & 0 \end{array} \to \begin{array}{cc|c} 1 & 1 & 0\\ 0 & 0 & 0 \end{array}$

Solving this simple equation yields a possible eigenvector $\overrightarrow{v_{2}} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$ .

Let $\begin{bmatrix} G_{\infty} \\ L_{\infty} \end{bmatrix}$ be the wizard's gold and lead supply once the spell is applied infinitely many times. In addition, the given spell matrix can be diagonalized as $\begin{bmatrix} 0.975 & 0.4 \\ 0.025 & 0.6 \end{bmatrix}^{t} = \begin{bmatrix} 16 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 1^{t} & 0 \\ 0 & 0.575^{t} \end{bmatrix} \begin{bmatrix} 16 & 1 \\ 1 & -1 \end{bmatrix}^{-1}$ .

Thus, $\begin{bmatrix} G_{\infty} \\ L_{\infty} \end{bmatrix} = \lim_{t \to \infty} \begin{bmatrix} 16 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 1^{t} & 0 \\ 0 & 0.575^{t} \end{bmatrix} \begin{bmatrix} 16 & 1 \\ 1 & -1 \end{bmatrix}^{-1} \begin{bmatrix} G_{0} \\ L_{0} \end{bmatrix} = \begin{bmatrix} 16 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \frac{1}{17} & \frac{1}{17} \\ \frac{1}{17} & -\frac{16}{17} \end{bmatrix} \begin{bmatrix} G_{0} \\ L_{0} \end{bmatrix}$ .

Solving the above equation for $G_{\infty}$ and $L_{\infty}$ yields $G_{\infty} = \frac{16}{17}(G_{0} + L_{0})$ and $L_{\infty} = \frac{1}{17}(G_{0} + L_{0})$ .

Therefore, $G_{\infty} : L_{\infty} = \frac{16}{1}$ meaning that $p = 16, q = 1$ and $p + q = \boxed{17}$

Let $G_{\infty}$ and $L_{\infty}$ be the values of $G$ and $L$ after infinitely many iterations. Assuming that the values converge, we have:

$G_\infty = 0.975 G_\infty + 0.4 L_\infty \\ L_\infty = 0.025 G_\infty + 0.6 L_\infty$

Dividing by $L_{\infty}$ :

$\frac{G_\infty}{L_\infty} = 0.975 \frac{G_\infty}{L_\infty} + 0.4 \\ 1 = 0.025 \frac{G_\infty}{L_\infty} + 0.6$

Finally:

$\frac{G_\infty}{L_\infty} = \frac{0.4}{1 - 0.975} = \frac{1 - 0.6}{0.025} = 16$