Given: A, B ∈ Mn(F) and C, D ∈ Mm(F)
A and B are equivalent, so are C and D.
Im not sure how to approach this.
But since A is equivalent to B and C is equivalent to D, there exist P, Q ∈ Mn(F) and S, T ∈ Mm(F)such that A = PBQ and C = SDT.
To prove that A⊕C is equivalent to B⊕D, do I show that there exist an X, Y ∈ M(m+n)(F) such that A⊕C = X(B⊕D)Y?
If so, how?
If I understand your notation:
$$A\oplus C=\left[\begin{matrix}A&0\\0&C\end{matrix}\right], B\oplus D=\left[\begin{matrix}B&0\\0&D\end{matrix}\right]$$
then you can certainly do that. The solution is:
$$X=\left[\begin{matrix}P&0\\0&S\end{matrix}\right]=P\oplus S$$
and
$$Y=\left[\begin{matrix}Q&0\\0&T\end{matrix}\right]=Q\oplus T$$