<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] By inspection, the vectors $(1,-1,1,-1,1,-1,1,-1)$, $(1,0,-1,0,1,0,-1,0)$, and $(0,1,0,-1,0,1,0,-1)$ are all in the nullspace, and are linearly independent, so the rank is at most $5$. 
</s>