Elimination All the Way Down

Elimination finishes at an augmented matrix like

from which the solution $x = 1,\ y = 3$ can be read straight off. Carrying elimination all the way to a shape like that is called putting the matrix in reduced row echelon form (RREF). A matrix is in RREF when:

1. In every row that isn't all zeros, the first nonzero entry is a $1$. Call it that row's pivot.
2. Each pivot sits further to the right than the pivot in the row above it.
3. A pivot is the only nonzero entry in its column.
4. Any rows of all zeros sit at the bottom.

Every matrix has exactly one RREF, and the three row operations — swap two rows, scale a row by a nonzero number, add a multiple of one row to another — are all you need to reach it.

Task: write rref(matrix). It takes a list of rows (each a list of numbers, with at least one row and one column, any shape) and returns its RREF as a new list of rows.

• Treat any value whose absolute size is below $10^{-9}$ as zero when looking for a pivot — floating-point arithmetic leaves tiny leftovers behind.
• Round every entry to 4 decimal places.

Once a system's augmented matrix is in RREF, the chapter's three cases are visible at a glance: a row reading $[\,0 \;\cdots\; 0 \mid 1\,]$ means no solution; a column of unknowns with no pivot means a free unknown and infinitely many solutions; otherwise the last column is the one and only answer.