Code:
$ rank(PQ)+rank(QR)=rank\begin{pmatrix}PQ & 0\\ 0 & QR\end{pmatrix}\leq rank\begin{pmatrix}PQ & 0\\ Q & QR\end{pmatrix}=rank\begin{pmatrix}0 & PQR\\ Q & 0\end{pmatrix}=r(PQR)+r(Q) $
Code:
$ rank(PQ)+rank(QR)=rank\begin{pmatrix}PQ & 0\\ 0 & QR\end{pmatrix}\leq rank\begin{pmatrix}PQ & 0\\ Q & QR\end{pmatrix}=rank\begin{pmatrix}0 & PQR\\ Q & 0\end{pmatrix}=r(PQR)+r(Q) $
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