Frederick - beautiful! I take it back: people should not take powers of the matrix

\[ \left( \begin{array}{cc} 1 & 1 \\\\ 1 & 0 \end{array} \right) \]

by hand, they should take powers of the matrix

\[ \left( \begin{array}{cc} f & g \\\\ h & 0 \end{array} \right) \]

by hand. The paths we're trying to count will pop out even more visibly. But one still needs to think, to understand exactly why it works! It's not magic, just logic.

\[ \left( \begin{array}{cc} 1 & 1 \\\\ 1 & 0 \end{array} \right) \]

by hand, they should take powers of the matrix

\[ \left( \begin{array}{cc} f & g \\\\ h & 0 \end{array} \right) \]

by hand. The paths we're trying to count will pop out even more visibly. But one still needs to think, to understand exactly why it works! It's not magic, just logic.