I am to prove or find a counterexample for the following problem: Let $A, B, C, D, F \in GL(n, \mathbb{R})$. If $D^{-1}(A+B+C)D=F$ is correct then $A=F-B-C$ is correct.
I have not found a counterexample though there might be one - I don't know. This is how far I get:
Let $E=A+B+C$. Then the problem can be expressed as "If $D^{-1}ED=F$ is correct then $E=F$ is correct." Also, $D^{-1}ED=F \Leftrightarrow ED=DF$
But I have no clue how I could show $ED=DF \Rightarrow E=F$.