Reduced Row-Echelon Form

1. The first nonzero number in a row is a 1. (We call it a leading 1).
2. All rows of zeros (if there are any) are together at the bottom of the matrix.
3. Each column that contains a leading 1, has only zeros below it.  In other words, for any two successive non-zero rows, the leading 1 in the lower row is farther to the right than the leading 1 in the higher row.

A matrix is said to be in Reduced-Row-Echelon-Form if these three conditions plus one more are satisfied:

4. Each column that contains a leading 1 has zeros everywhere else.

The idea here is that once the matrix is in one of these forms, the system of equations that has this as its augmented matrix can be solved with little or almost no effort.

Reduced-Row-Echelon-Form is also known as Gauss-Jordan Elimination.

RREF Reduction Procedure

The RREF reduction procedure is carried out using simple row operations.  It can be used to solve simultaneous linear equations, where an augmented matrix is used to represent the coefficients and constants.  (See "augmented matrix")

Simple row operations are

Start by multiplying row 1 by a scalar so that a₁₁ is 1.  That is, multiply row 1 by 1/a₁₁.

Then eliminate all the non-zero values in the first column by adding an appropriate multiple of the first row to each of the other rows, ignoring for now the effect on the cells to the right of the first column.

After doing these two steps, the first column will contain 1, followed by all zeros.

Now change the leading cell of row two to 1 the same way -- by multiplying it by an appropriate scalar.  Change all the other cells in column 2 to zero the same way as you did the first column -- by adding an appropriate multiple of the first row to each of the other rows.  Notice that you can't "mess up" the first column by doing this, since row two has a zero in column 1.

Continue this way down the diagonal of the matrix, setting all the cells to 1, and all the cells above and below that cell to zero.  If you come to a cell that is already zero, interchange its row with one below it, and continue -- whenever this happens, you'll find that you'll have a row of zeros at the end, meaning your matrix is not of "full rank".

When you're all done, you'll have a matrix that represents the "solved" form of a set of linear equations.

Related Pages in this website

Definitions - includes a definition of augmented matrix, which is a way of representing simultaneous linear equations.  Then these equations can be solved using RREF, explained above, on this page.

Cramer's Rule

Vectors -- the "dot" product and the "cross" product, explained.

The webmaster and author of the Math Help site is Graeme McRae.
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