Let $$ Q=\begin{bmatrix}\phantom{-}A&B\\-B&A\end{bmatrix} $$ be a block matrix where $A,B\in \mathbb{R}^{n\times n}$. Prove that $$\mathrm{Rank}(Q)=2\mathrm{Rank}(\left[\,A \ \: B\,\right]).$$

In other words, I want to show that the rank of $Q$ is twice the rank of the matrix $Q_U:=[A \: \: B]\in \mathbb{R}^{n\times 2n}$. This equation holds when $A$ and $B$ are Invertible. I was wondering if this also holds for any matrix $A$ and $B$.

Note that $A$ and $B$ do not commute. However, we can assume that $A$ and $B$ are symmetric. Any comment or response is appreciated.


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