# Rank of block matrix with equal diagonals

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.