GAUSSJORDAN ELIMINATION Dr Shildneck Fall 2015 AUGMENTED MATRICES

GAUSS-JORDAN ELIMINATION Dr. Shildneck Fall, 2015

AUGMENTED MATRICES

What is an Augmented Matrix? • A matrix that has “extra” columns added to it • The extra columns represent something that relates to what you are trying to find • There are two typical augmentations that we will use. • Augmentation for solving systems (Today) • Augmentation for finding inverses of matrices (Monday)

Augmenting Matrices for Solving Systems of Equations • Write the coefficient matrix using the left bracket only. • Draw a “dotted” line on the right side of the last column • Add a new column to the right using the numbers from the “arguments” • Add the right bracket

Example 1 – To Solve a System [ 3 -2 5 1 ] 13 26

Example 2 – To Solve a System [ 1 1 1 2 -1 3 1 -1 5 6 9 14 ]

Augmenting to find Inverses of Matrices • The key to finding an inverse using augmented matrices is to recall the definition of an inverse matrix. • The inverse of a matrix A is that matrix A-1 that when multiplied with A gives you the identity matrix [ I ] • The goal then is to augment the matrix with the result that you want after multiplying (the identity). • If you think about it, this is the same thing we did with systems.

Example 3 – Finding an Inverse To find the inverse of: Augment the matrix like: [ 5 6 2 2 ] 1 0 0 1

Example 4 – Finding an Inverse To find the inverse of: Augment the matrix like: [ 1 2 0 3 -1 1 2 2 4 ] 1 0 0 0 1

Solving Systems Using Augmented Matrices Gauss-Jordan Elimination

Gauss-Jordan Elimination • The goal of Gauss-Jordan Elimination is to use multiplication and addition to change the coefficient part of the augmented matrix into the identity matrix. • Anything you do, you do to the entire row. • Anything you do an individual row changes the entire row. • Anything you do to a combination of rows is stored at the last row you mention in the step.

Gauss-Jordan Elimination Things that you can do to change rows: • Multiply a row by a number (divide) • Add two rows together (subtract) • Multiply a row and add it to another row.

Example 1 – Solve the System: [ ][ ][ ] 3 -2 5 1 13 26 2 R 2 + R 1 13 0 5 1 65 26 1 0 5 1 -5 R 1 + R 2 1 0 0 1 5 26 5 1

Example 2 – Solve the system: [ 1 1 1 2 -1 3 1 -1 5 6 9 14 ] The solution to this system is (1, 2, 3).

ASSIGNMENT Assignment #9 – Solving Systems Using Gauss-Jordan