MAT 2401 Linear Algebra 1.1, 1.2 Part I Gauss- Jordan Elimination

HW Written Homework

Time Part I may be a bit longer today. Part II will be shorter next time.

Preview Introduce the Matrix notations. Study the Elementary Row Operations. Study the Gauss-Jordan Elimination.

Example 1 Elimination:

Example 1 Elimination:Geometric Meaning

How many solutions? Q: Given a system of 2 equations in 2 unknowns, how many solutions are possible? A:

How many solutions? Q: Given a system of 3 equations in 3 unknowns, how many solutions are possible? A:

How many solutions? Q: Given a system of 3 equations in 3 unknowns, how many solutions are possible? ______ System

Unique Solution We will focus only on systems of unique solution in part I. Such systems appear a lot in applications.

Example 2 Elimination:

Observation 1 Q: Why eliminations are not good? A:

Observation 2 Compare the 2 systems: Q: Are the 2 systems the same? A:

Observation 2 Compare the 2 systems: Q: What do the 2 systems have in common? A:

Observation 2 Compare the 2 systems:

Observation 2 Compare the 2 systems: Q: Which system is easier to solve? A:

Extreme Makeover? We want a solution method that it is systematic, extendable, and easy to automate it can transform a complicated system into a simple system

Extreme Makeover? We want a solution method that it is systematic, extendable, and easy to automate it can transform a complicated system into a simple system

Extreme Makeover? We want a solution method that it is systematic, extendable, and easy to automate it can transform a complicated system into a simple system

Extreme Makeover? We want a solution method that it is systematic, extendable, and easy to automate it can transform a complicated system into a simple system

Gauss-Jordan Elimination

Before we can describe our systematic solution method, we need the matrix notations.

Essential Information A system can be represented compactly by a “table” of numbers.

Matrix A matrix is a rectangular array of numbers. If a matrix has m rows and n columns, then the size of the matrix is said to be mxn.

Example 2 Write down the (Augmented) matrix representation of the given system.

Coefficient Matrix The left side of the Augmented matrix is called the Coefficient Matrix.

Elementary Row Operations We can perform the following operations on the matrix 1. Switching 2 rows. 2. Multiplying a row by a constant. 3. Adding a multiple of one row to another.

Elementary Row Operations We can perform the following operations on the matrix 1. Switching 2 rows.

Elementary Row Operations We can perform the following operations on the matrix 2. Multiplying a row by a constant.

Elementary Row Operations We can perform the following operations on the matrix 3. Adding a multiple of one row to another.

Elementary Row Operations Theory: We can use the operations to simplify the system without changing the solution. 1. Switching 2 rows. 2. Multiplying a row by a constant. 3. Adding a multiple of one row to another.

Elementary Row Operations Notations (examples) 1. Switching 2 rows. 2. Multiplying a row by a constant. 3. Adding a multiple of one row to another.

Gauss-Jordan Elimination Main Idea: We want to use elementary row operations to get the matrix into the form (reduced row-echelon form RREF)

Gauss-Jordan Elimination Main Idea: We want to use elementary row operations to get the matrix into the form (reduced row-echelon form RREF) The order of creating “0” and “1” is extremely important! 123

Example 2 123

Remarks Notice sometimes 2 “parallel” row operations can be done in the same step. The procedure (algorithm) is designed so that the exact order of creating the “0”s and “1”s is important.

Remarks Try to avoid fractions!!

How do I Confirm My Answers?

Example 3 Use Gauss-Jordan Elimination to solve the system.

Example 3 123