$B$ is similar to $C$ if and only if $A\oplus B$ is similar to $A\oplus C$ Let $A\in M_m(\mathbb{C})$ and $B,C\in M_n(\mathbb{C})$.
Let $$ A\oplus B:=
\begin{pmatrix}
A & 0 \\
0 & B
\end{pmatrix}$$ Show that the following are equivalent:
(i) $B$ is similar to $C$.
(ii) $A\oplus B$ is similar to $A\oplus C$.
Hope you can help! Thanks!
 A: Let $A\oplus B$ denote the block diagonal matrix with matrices $A$ and $B$ as diagonal entries.
Assume $B\sim C$. Consider $A\oplus B$ and $A\oplus C$. Because $B\sim C$, there exists a sequence of elementary row opperations that change $B$ into $C$. Applying these same opperations in the same order to the lower section of $A\oplus B$ (the $B$ block), still turns $B$ into $C$ and so yields $A\oplus C$.
Assume $A\oplus B\sim A\oplus C$. Then there exists a sequence of elementary row opperations that change $A\oplus B$ into $A\oplus C$. Notice that since the top section (the $A$ block) stays the same, it follows that WLOG we can assume we only modified the $B$ block section, since adding a scalar multiple of a row of $A$ to a lower row will be undone at some point to return to the block diagonal form. But then this same sequence of moves transforms $B$ into $C$.
A: Calculate
$$
\begin{bmatrix} S_1 & S_2 \\ S_3 & S_4 \end{bmatrix} · \begin{bmatrix} T_1 & T_2 \\ T_3 & T_4 \end{bmatrix} = \begin{bmatrix} S_1 T_1  + S_2 T_3 & S_1 T_2 + S_2 T_4 \\ S_3 T_1 + S_3 T_3 & S_2 T_2 + S_4 T_4 \end{bmatrix}.
$$
This result holds as long as the dimensions involved fit together. This can be calculated entry-wise. So
$$
\begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix} · \begin{bmatrix} T_1 & T_2 \\ T_3 & T_4 \end{bmatrix} = \begin{bmatrix} A T_1 & A T_2 \\ B T_3 & B T_4 \end{bmatrix},
$$
and similarly
$$
\begin{bmatrix} S_1 & S_2 \\ S_3 & S_4 \end{bmatrix} · \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix} = \begin{bmatrix} S_1 A & S_2 B \\ S_3 A & S_4 B\end{bmatrix}.
$$
