<s>[INST] How to show the matrix has Rank $\le 5$ 
I want to show that the following matrix has Rank $\le 5$.

The matrix is 
\begin{bmatrix}
2&1&1&1&0&1&1&1\\
1&2&1&1&1&0&1&1\\
1&1&2&1&1&1&0&1\\
1&1&1&2&1&1&1&0\\
0&1&1&1&2&1&1&1\\
1&0&1&1&1&2&1&1\\
1&1&0&1&1&1&2&1\\
1&1&1&0&1&1&1&2
\end{bmatrix}
I found that there is a submatrix in the matrix which has rank $ =4$ given by
$[2,1,1,1],[1,2,1,1],[1,1,2,1],[1,1,1,2]$.
I need to show the given matrix has at least 3 zero rows in order to show that Rank $\le 5$..
But I dont know how to show it.
Can someone help.
 [/INST] The given matrix is equal to $A+B$ where
$$
A=\begin{bmatrix}
1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1
\end{bmatrix},\quad
B=\begin{bmatrix}
1&0&0&0&-1&0&0&0&\\
0&1&0&0&0&-1&0&0&\\
0&0&1&0&0&0&-1&0&\\
0&0&0&1&0&0&0&-1&\\
-1&0&0&0&1&0&0&0&\\
0&-1&0&0&0&1&0&0&\\
0&0&-1&0&0&0&1&0&\\
0&0&0&-1&0&0&0&1&
\end{bmatrix}.
$$
These matrices have rank $1$ and $4$ respectively, so their sum has rank at most $1+4$.
</s>