What is the flaw in this Gaussian elimination?

Question

$\newenvironment{sysmatrix}[1] {\left[\begin{array}{@{}#1@{}}} {\end{array}\right]}$

I was tasked with getting this to reduced row-echelon form:

$$\begin{sysmatrix}{cccc|c} 1 & 3 & 1 & 1 & -1 \\ -2 & -6 & -1 & 0 & 1 \\ 1 & 3 & 2 & 3 & 2 \end{sysmatrix}$$

To get everything below $(1, 1)$ to 0, I went with $2r_1 + r_2 \to r_2$ and $-r_1 + r_3 \to r_3$:

$$\begin{sysmatrix}{cccc|c} 1 & 3 & 1 & 1 & -1 \\ 0 & 0 & 1 & 2 & -1 \\ 0 & 0 & 1 & 2 & 3 \end{sysmatrix}$$

Then below $(2, 3)$ to 0, $-r_2 + r_3 \to r_3$:

$$\begin{sysmatrix}{cccc|c} 1 & 3 & 1 & 1 & -1 \\ 0 & 0 & 1 & 2 & -1 \\ 0 & 0 & 0 & 0 & 4 \end{sysmatrix}$$

Then to fix the 1 above $(2, 3)$, $-r_2 + r_1 \to r_1$:

$$\begin{sysmatrix}{cccc|c} 1 & 3 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & -1 \\ 0 & 0 & 0 & 0 & 4 \end{sysmatrix}$$

This looks like reduced row-echelon to me. Yet I know that this isn't correct because the calculator tells me the correct answer is:

$$\begin{sysmatrix}{cccc|c} 1 & 3 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{sysmatrix}$$

Yet I genuinely cannot figure out where I went wrong here. The raw computations I've double checked, and each operation I performed here was a multiple of one row added to another, which I understand is legal. Clearly one of these is wrong, but how?

Answer 1

There are two rules.

1. In each row, the left most nonzero entry must be $1$.
2. Each column that contains $1$ as the left most nonzero in its row must have other entries zeroed in that column.

Your calculations are correct. Finish up by multiplying third row with $1/4$ and then add it to second row.