Rank of a matrix summation Assume $M\in\{0,1\}^{n\times n}$ of rank $r$. Is there an example of $M$ such that $rk(M+M')/rk(M)> 2$? Is there an example of $M$ such that $rk(M)/rk(M+M')> 2$?
 A: Due to the rank inequalities, the answer to the first question is no:
$$
rank(M+M') \le rank(M) + rank(M') = 2 rank(M).
$$
Here is an example of an $8\times 8$ matrix $M$ such that $rank(M)=8$, $rank(M+M^T)=2$:
$$
M:=
\left(
\begin{array}{cc|cc|cc|cc}
     0&     0&     1&     0&     0&     1&     1&     0\\
     1&     0&     0&     0&     0&     0&     0&     1\\
     \hline
     0&     0&     0&     1&     0&     0&     0&     0\\
     0&     1&     0&     0&     0&     0&     1&     0\\
     \hline
     0&     1&     1&     0&     0&     0&     1&     0\\
     0&     0&     0&     1&     1&     0&     0&     0\\
     \hline
     0&     0&     0&     0&     0&     0&     0&     1\\
     0&     0&     1&     0&     0&     1&     0&     0\\
     \end{array}
     \right),
     M+M^T=
\left(
\begin{array}{cc|cc|cc|cc}
     0&     1&     1&     0&     0&     1&     1&     0\\
     1&     0&     0&     1&     1&     0&     0&     1\\
     \hline
     1&     0&     0&     1&     1&     0&     0&     1\\
     0&     1&     1&     0&     0&     1&     1&     0\\
     \hline
     0&     1&     1&     0&     0&     1&     1&     0\\
     1&     0&     0&     1&     1&     0&     0&     1\\
     \hline
     1&     0&     0&     1&     1&     0&     0&     1\\
     0&     1&     1&     0&     0&     1&     1&     0\\
     \end{array}
     \right).
$$
(Matlab did the rank computation). I could not find a smaller example.
