Question

I wonder if the Prof is correct in Lecture 2 at minute 8:19 where he said that the 3rd pivot is 5. I think that it is -3

Answer 1

The matrix after the first step of elimination looks like:
\[\left[\begin{matrix}1 & 2 & 1\\ 0 & 2 & -2\\ 0 & 4 & 1\end{matrix}\right]\]

We want to eliminate element 32 (that is 4). To do that, we multiply second row by 2 (to get 4 in 22 position):

\[\left[\begin{matrix}1 & 2 & 1\\ 0 & 4 & -4\\ 0 & 4 & 1\end{matrix}\right]\]

Finally, we subtract second line from the third:
\[\left[\begin{matrix}1 & 2 & 1\\ 0 & 2 & -2\\ 0-0 & 4-4 & 1-(-4)\end{matrix}\right] =\left[\begin{matrix}1 & 2 & 1\\ 0 & 2 & -2\\ 0& 0& 5\end{matrix}\right]. \]