This is a partial answer, now with added material. Assume that $n$ is a multiple of 3, the other two cases should be similar. I'll denote $C_1\mapsto a$ and $C_2\mapsto b$, such that for, e.g., $n=6$, $$\tag{1}\label{1} A = \begin{pmatrix} a_1 & 1 & 0 & 0 & 0 & 0 \\ a_2 & 0 & 1 & 0 & 0 & 0 \\ a_3 &...

Let $D\subset \Bbb{R}$ be a dense set. Question: Does $D$ intersect every closed uncountable sets? The answer is negative. As the Cantor set $\mathcal{C}$ is n.w.dense , it's exterior $\Bbb{R}\setminus \overline{\mathcal{C}}=\Bbb{R}\setminus {\mathcal{C}}$ is dense in $\Bbb{R}$ but it doesn't i...