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If Given the transition matrices:

$$ P_{A \rightarrow B} \begin{bmatrix} 3 &1 \\ 5 & 2 \end{bmatrix} P_{B \rightarrow C} \begin{bmatrix} 7 &2 \\ 4 & -1 \end{bmatrix} $$ How would one determine the transition matrix of A to C? My first intuition was to premultiply the second to the first. Any help would be appreciated.

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Usually the action of a matrix on a vector is represented as a left multiplication (rows by columns) of the matrix by the vector: $P \vec v$ . In this case you are right: we have $$ P_{A\to C}=P_{B\to C}P_{A\to B} $$

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