<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] Call column $j$ of this matrix $C_j$, and denote by $\def\2{\mathbf 2}\2$ the column vector with all $8$ entries equal to $2$. Note that for $j=1,2,3,4$ one has $C_j+C_{j+4}=\2$. Thus $C_6=\2-C_2=C_1+C_5-C_2$, and similarly $C_7=C_1+C_5-C_3$ and $C_8=C_1+C_5-C_4$. The last three columns being in the span of the first five, the rank of the matrix is at most$~5$.
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