Gaussian elimination and pivots

Question

I'm having trouble answering the question posed below ("Question: Which x on the left...").

It seems like there's only 2 options as the first two along the diagonal are already boldfaced. I'm not sure which one and why though.

Answer 1

Short answer:

The $x$ on line 3, column 3, since that is the pivot you will use on next step (provided that is non zero)

Long "picky" one:

The problem is maybe in the usage of the letter $x$ for the whole matrix, since in fact we are considering a matrix of real numbers, which of course can be different, so naming all of them $x$ is weird.

That being said the question is still "hard" to answer. As you have already noticed there are only 2 options, the $x$'s on the third row. The first thing is, we have to suppose that the matrix has maximal rank, so at least one of those $x$'s is different from zero. The second one is that we can choose any $x\neq 0$, but then, without some transformation, the final matrix might not be triangular, that is, if you choose the $x$ on the 4th column you would get: $$ \left[\begin{array}{cc} x & \boldsymbol{x} \\ x & x \\ \end{array}\right] \longrightarrow \left[\begin{array}{cc} x & \boldsymbol{x} \\ \boldsymbol{x} & \\ \end{array}\right] $$

So that the full matrix ends being $$ \left[\begin{array}{cccc} \boldsymbol{x} & x & x & x \\ & \boldsymbol{x} & x & x \\ & & x & \boldsymbol{x} \\ & & \boldsymbol{x} \\ \end{array}\right] $$ but using a column transformation, which gives an equivalent system you would still get a triangular one. Moreover you can already use this matrix to solve it, but starting with the 3rd variable, not the 4th.