Suppose I have matrix
$B:= \begin{bmatrix}4 & -2 & 2\\-2 & 5 & 3\\ 2 & 3 & 7 \end{bmatrix} $
Performing Gaussian Elimination we get:
EDIT corrected mistake. I mistakenly flipped the sign for for the top right element. Supposed to be +2 instead of -2.
Step 1 -
~$\begin{bmatrix}4 & -4 & 2\\0 & 4 & 4\\ 0 & 4 & 6 \end{bmatrix} $
Step 2
~$\begin{bmatrix}4 & -4 & 2\\0 & 4 & 4\\ 0 & 0 & 2 \end{bmatrix} $
Can somebody explain to me how step 1 was done? It's not obvious to me what the steps were. I would appreciate it if somebody could break it down.
Something went wrong here. What they've tried to do -- judging from the results in the first two columns -- is $R_2\Rightarrow R_2+R_1/2$ and $R_3\Rightarrow R_3-R_1/2$, but when I do that, I get $$\begin{bmatrix}4 & -2 & -2\\0 & 4 & 2\\ 0 & 4 & 8 \end{bmatrix}$$
Perhaps the first row was in fact $(4, -2, 2)$.
But that doesn't explain what's happened to the first row, either, which shouldn't change in any application of gaussian elimination that doesn't require row swapping (this one does not).
Completing the elimination, I then do $R_3\Rightarrow R_3-R_2$, to get
$$\begin{bmatrix}4 & -2 & -2\\0 & 4 & 2\\ 0 & 0 & 6 \end{bmatrix}$$