Aim: put any matrix into [[REF]] using [[EROs]].

== Algorithm ==

  - Re-order the rows so that the leftmost [[leading entry]] in the matrix is in the top row. 
  - Divide all of the top row by its leading entry, so that this entry becomes a $1$.
  - "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 $0$.
  - 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:

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