<s>[INST] Matrix gauss continuation Suppose we have the matrix $4\times4$, $R_1 =(0,0,1,1)$, $R_2=(0,1,1,0)$, $R_3=(1,1,0,0)$, $R_4(1,0,0,0)$ and suppose we want to see the rank.The difficult part is that you cannot order them somehow to use gauss.Any ideas what should be done in such situations?
 [/INST] You can subtract any row from any other without changing the rank:
$$\begin{pmatrix}0&0&1&1\\0&1&1&0\\1&1&0&0\\1&0&0&0\end{pmatrix}\sim\begin{pmatrix}0&0&1&1\\0&1&1&0\\0&1&0&0\\1&0&0&0\end{pmatrix}\sim\begin{pmatrix}0&0&1&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{pmatrix}\sim\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{pmatrix}$$
So the rank is full.
You can also reorder the rows at the beginning as this doesn't change the rank:
$$\begin{pmatrix}1&0&0&0\\1&1&0&0\\0&1&1&0\\0&0&1&1\end{pmatrix}$$
and continue as before.
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