Question about row operations and row-echelon form,

Question

If I have a matrix, with, say, the first two columns consisting of all zeroes, then is the first entry of the third column, which is non-zero, my first pivot variable, so that when solving Ax=b, for x=$(x_1, x_2, ..., x_n)$, $x_1$ and $x_2$ are free variables.

I'm a little confused, since I am used to seeing my pivot variables on the main diagonal; in the above situation described, my first (and subsequent) pivot variables are all above the main diagonal, because of the first two columns of zeroes.

Thanks,

Answer 1

The short answer is yes. If your matrix looks like, say,

\begin{pmatrix} 0 & 0 & 5 & 1 & 7 \\ 0 & 0 & 2 & -3 & 4 \\ 0 & 0 & 0 & -7 & 1 \end{pmatrix}

then pivoting on $5$ is perfectly acceptable.