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:

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.