Given two change of basis matrices, $A>B$ and $B>C$ find the matrix $C>A$
I'm confused about how change of basis matrices work when combining them. I know you can find $B>A$ by finding the inverse matrix, but how can you combine two change of basis matrices? Is it as simple as finding the two inverse matrices then multiplying them together?
You are correct.
Let $A,B,$ and $C$ be bases. Let $[I]_A^B$ be the change of basis matrix from basis $A$ to $B$.
Then $[I]_A^C = [I]_B^C \; [I]_A^B$ (changing from $A$ to $B$ and then from $B$ to $C$ yields a change from $A$ to $C$).
A consequence of this is that $[I]_A^B [I]_B^A = [I]_B^B =$ identity matrix (changing from $B$ to itself does nothing). Therefore, $[I]_B^A = \left([I]_A^B\right)^{-1}$.
Finally, $[I]_C^A = \left([I]_A^C\right)^{-1} = \left([I]_B^C[I]_A^B\right)^{-1} = \left([I]_A^B\right)^{-1}\left([I]_B^C\right)^{-1}$.