Elimination Matricies

For matrix A, we want to make a series of matrix M which will zero columns out. This is a algorithms approach for doing this, which is also applied columnwise.
nicely, we can undo our operations
and you can compose them by subtracting together
*THIS IS ONLY TRUE when we are applying in the right ordering, row 1 to row 2, etc.
pivoting this procedure breaks on:

\begin{equation} A = \mqty(0 & 4 \\ 4 & 9) \end{equation}

because it requires dividing by \frac{1}{0}; to “fix” this, reorder the equatinos.
you can flip both rows and columns.
partial pivoting: swap rows to use the largest magnitue element in the column under consideration full pivoting: swap rows and columns to use the largest element on the upper left Remember to change the order of b in the first case, and both b (rows) and c (columns) in the second case.