# Find a $3\times3$ matrix whose reduced row echelon form has two leading ones and whose row space intersects its column space by the line $x_1=x_2=x_3$

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.

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 $$\pmatrix{ 1&1&1\\ 1&0&1\\ 1&0&1 }$$