A is a $2 \times 2$ matrix. Show that there exist $2 \times 2$ matrices with certain rank requirements.

Question

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!

Let $C$ be your favorite rank 1 $2\times2$ matrix, let $D$ be your favorite rank 2 $2\times2$ matrix, let $B_1=C-A$, let $B_2=D-A$. — Gerry Myerson, Sep 11 '14 at 07:14
@GerryMyerson I had the same thought - if you want to post yours as an answer I will gladly delete mine. — Zubin Mukerjee, Sep 11 '14 at 07:15
@Zubin, please don't delete your answer on account of my comment. — Gerry Myerson, Sep 11 '14 at 07:16

score 1 · Answer 1 · answered Sep 11 '14 at 07:14

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.

A is a $2 \times 2$ matrix. Show that there exist $2 \times 2$ matrices with certain rank requirements.

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