If $A=B+C$ and $\operatorname{rank} A=2$, $\operatorname{rank} B=\operatorname{rank} C=1$, are $B$ and $C$ unique?

Question

The question is in the title:

If $A$ is rank $2$ and $A=B+C$ with $B,C$ having rank $1$ each, then are $B$ and $C$ unique? (up to ordering of course)

I hesitate to say that this is true since a counterexample isn't immediately coming to mind, and I know that $A=\sigma_1u_1v_1^T+\sigma_2u_1v_1^T$ where $u_i,v_i$ are pulled from the SVD of $A$, but I can't come up with a proof.

Any help is appreciated.

Answer 1

$\begin{bmatrix}2&0\\0&2\end{bmatrix} = \begin{bmatrix}1&1\\1&1\end{bmatrix}+\begin{bmatrix}1&-1\\-1&1\end{bmatrix} = \begin{bmatrix}2&0\\0&0\end{bmatrix}+\begin{bmatrix}0&0\\0&2\end{bmatrix}$