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I am confused on how a matrix can exist

I tried doing something like this $$ \begin{bmatrix}1& 0& 1\\0& 1& 1\\0& 0& 0\end{bmatrix} $$

but this only intersects with $x_1=x_2$ and not with $x_3$

I just don't see how a $3\times3$ matrix reduced to only two leading $1$'s can produce that result.

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Note that the matrix we want doesn't need to be (and, in fact, can't be) in reduced row echelon form. All we need is any $3 \times 3$ matrix of rank $2$ with the described row-space/column-space property.

For example, we can take the matrix $$ \pmatrix{ 1&1&1\\ 1&0&1\\ 1&0&1 } $$

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