The Gauss-Jordan Elimination Algorithm

Definitions

The Algorithm

Solutions of Linear Systems

Answering Existence and Uniqueness questions

The Gauss-Jordan Elimination Algorithm

Solving Systems of Real Linear Equations

A. Havens

Department of Mathematics University of Massachusetts, Amherst

January 24, 2018

A. Havens

The Gauss-Jordan Elimination Algorithm

Definitions

The Algorithm

Solutions of Linear Systems

Answering Existence and Uniqueness questions

Outline

1 Definitions Echelon Forms Row Operations Pivots

2 The Algorithm Description The algorithm in practice

3 Solutions of Linear Systems Interpreting RREF of an Augmented Matrix The 2-variable case: complete solution

4 Answering Existence and Uniqueness questions The Big Questions Three dimensional systems

A. Havens

The Gauss-Jordan Elimination Algorithm

Definitions

The Algorithm

Solutions of Linear Systems

Echelon Forms

Row Echelon Form

Answering Existence and Uniqueness questions

Definition A matrix A is said to be in row echelon form if the following conditions hold

1 all of the rows containing nonzero entries sit above any rows whose entries are all zero,

2 the first nonzero entry of any row, called the leading entry of that row, is positioned to the right of the leading entry of the row above it,

Observe: the above properties imply also that all entries of a column lying below the leading entry of some row are zero.

A. Havens

The Gauss-Jordan Elimination Algorithm

Definitions

The Algorithm

Solutions of Linear Systems

Echelon Forms

Row Echelon Form

Answering Existence and Uniqueness questions

Such a matrix might look like this:

a

0

0

b 0

0

c

,

00000d

where a, b, c, d R? are nonzero reals giving the leading entries, and `' means an entry can be an arbitrary real number.

Note the staircase-like appearance hence the word echelon (from french, for ladder/grade/tier).

Also note that not every column has a leading entry in this example.

A. Havens

The Gauss-Jordan Elimination Algorithm

Definitions

The Algorithm

Solutions of Linear Systems

Echelon Forms

Row Echelon Form

Answering Existence and Uniqueness questions

A square matrix in row echelon form is called an upper triangular

matrix.

E.g. 1 2 3 4

0 5 6 7

0

0

8

9

0 0 0 10

is a 4 ? 4 upper triangular matrix.

A. Havens

The Gauss-Jordan Elimination Algorithm

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