A is a $2 \times 2$ matrix. Show that there exist $2 \times 2$ matrices $B_1$ and $B_2$ such that $\textrm{rank}(A+B_1)=1$ and $\textrm{rank}(A+B_2)=2$.
I can write examples of $B_1$ and $B_2$, but could anyone give me a more thorough answer? Thank you very much!
Find any $2$ by $2$ matrix with rank $1$, call it $C$. Find any $2$ by $2$ matrix with rank $2$, call it $D$.
Now, for any $A$, you can let $B_1=C-A$ and $B_2=D-A$. These differences exist because the set of $2$ by $2$ matrices is closed under subtraction, so $B_1$ and $B_2$ exist.