Similar Invertible Matrices

Question

If $B$ is similar to $A$, show that there exists an invertible matrix $S$ and another marix $T$ such that $A=ST$ and $B=TS$.

So I know that $B$ being similar to $A$ implies $M^{-1}AM=B$. Can I just say let $A$ be a product of 2 invertible matrices $S$ and $T$ then manipulate $M$ to be either $S$ or $T$?

Answer 1

Let $A$ and $B$ be similar. Then we have $A=S(BS^{-1})=ST$ with $T=BS^{-1}$. It follows that $$ TS=BS^{-1}S=B. $$

Answer 2

Write the similarity condition as $AM=MB$.

Can you see, that you are quite close to the wanted product presentation?