### Row Echelon Form

##### 1. Row echelon form

We use row operations to simplify matrices. A natural question is: How simple can a matrix become by row operations?

Example The following matrix

 [ 1 4 0 5 ] 0 0 2 6 0 0 0 3

has the upside down staircase shape,

 [ * # # # ] 0 0 * # 0 0 0 *

in which * denotes nonzero numbers and # denotes any numbers. We claim that the shape cannot be improved by row operations.

First, the row operations r[row 1] + [row 2], r[row 1] + [row 3], r[row 2] + [row 3], with r ≠ 0, produce the following shapes.

 [ * # # # ] * # # # 0 0 0 *
 [ * # # # ] 0 0 * # * # # #
 [ * # # # ] 0 0 * # 0 0 * #

All are more complicated than the original one. Moreover, adding multiples of lower rows to upper rows, such as r[row 3] + [row 1], will give us the same shape.

Next, the operation [row 2] ↔ [row 3] gives us

 [ * # # # ] 0 0 0 * 0 0 * #

Instead of simplifying the shape, this makes the shape uglier. You may check out the effects of the other exchange operations.

Finally, multiplying a nonzero number to a row does not change the shape, so that the shape is still not improved.

The following summarizes the simplest shape one can get by row operations.

A row echelon form is characterized by two properties

• The shape cannot be further improved by row operations
• The rows are arranged from the "longest" to the "shortest"