So I have the bases $\displaystyle S=(1,0),(0,1) $ and $\displaystyle C=(3,1),(5,2)$

and a transformation matrix (in terms of S) $\displaystyle (0,1;1,0) $. To find this transformation matrix in terms of C I did this;

(basis S in terms of C)(transformation in terms of S)(basis C in terms of S)

I got matrix $\displaystyle (-13,21;8,13)$ if this is the right thing to do, how can I prove it?