MST10030 notes wiki

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Aim: put any matrix into REF using EROs.

Algorithm

1. Re-order the rows so that the leftmost leading entry in the matrix is in the top row.
2. Divide all of the top row by its leading entry, so that this entry becomes a $1$.
3. “Pivot about the leading 1”: subtract multiples of the top row from each row below so that all entries below the leading $1$ in the top row become $0$.
4. Go back to the start, ignoring the top row (until no rows remain, except possibly zero rows).

If we want, we can go further and put the matrix into RREF. First put it into REF as above, and then:

1. Look at the non-zero row nearest the bottom of the matrix
2. Pivot about the leading $1$ in that row and use it to make zeros above: subtract multiples of that row from each row above
3. Move to the next row up, and go to step 2 (until no rows remain).