Row Echelon Form

Row echelon form is a variant of an augmented matrix which can be used to solve a Linear System. They can also be used to check if a linear system is consistent.

Definition §

The leading entry is the leftmost non-zero entry of a row.

In order for a matrix to be in row-echelon form (REF), it must:

• Have all rows containing zeros only be at the bottom of the matrix
• More importantly, the leading entry of every row (also called the pivot) must be to the right of the leading entry of the row above

Non-unique

The REF of a linear system is non-unique, there are multiple different valid REFs.

These two conditions imply that all entries in a column below a leading entry must be zero. For example:

$$\left[\begin{array}{ccc}3& 2& 1\\ 0& -3& 5\\ 0& 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 2\end{array}\right]$$

are all matrices in REF.

Gaussian Elimination §

Gaussian elimination uses the elementary row operations in order to transform a matrix, $\mathbf{M}$, to REF.

Note that because elementary row operations do not affect the solutions of the system, the REF matrix can still be used to solve the system.

Example §

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