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MAT 1234Calculus I
9.4 System of Linear Equations: Matrices
http://myhome.spu.edu/lauw
HW and …
WebAssign 9.4 Quiz 3.8, 3.9
Important Announcement
We are 3 sections (4 periods) ahead of the other section.
No new materials tomorrow. Less materials to cover in the Exam 3. Please study over the weekend! If you
do well in the first 3 exams, you can opt out from the final exam.
Quiz 9
1 point curved is added to everyone.
Some of you have more than 10 points.
Be sure to study the solutions online.
One of the problems in this section will be in Exam 3.
Preview
Develop the Matrix notations. Study the Elementary Row
Operations. Study the Gaussian-Jordan
Elimination* (Easy version).
Preview
If you have learned these before, it is a good time to review them.
If you are planning to take Linear Algebra, you are going to see this in the early part of the course.
How does Maple do it?
2 5Solve
4 6
x y
x y
Example 1
Elimination:(Write this somewhere else. Do not erase this!)
2 5Solve
4 6
x y
x y
Example 1
Elimination: Geometric Meaning
2 5Solve
4 6
x y
x y
Equivalent Systems
Equivalent Systems Geometric Meaning
2 5
4 6
x y
x y
2 5Solve
4 6
x y
x y
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?
Unique Solution
We will focus only on systems of one solution because of the time constraints.
Such systems appear a lot in applications.
Example 2
Elimination:
4
Solve 2 2 5 11
4 6 8 24
x y z
x y z
x y z
Q & A
Q: Why eliminations are not good?
A:1.2.3.
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.
4 6 4 1 6
2 5 1 2 5
x y
x y
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.
1 2
1
2
n
m
Example 2
4
2 2 5 11
4 6 8 24
x y z
x y z
x y z
Write down the (Augmented) matrix representation of the given system.
Elementary Row Operations
We can perform the following operations on the matrix
1. Switching 2 rows.1 1 1 44
52 2 5 11 2 2 11
4 6 8 24 6 84 24
x y z
x y z
x y z
Elementary Row Operations
We can perform the following operations on the matrix
2. Multiplying a row by a constant.1 1 1 44
52 2 5 11 2 2 11
4 6 8 24 6 84 24
x y z
x y z
x y z
Elementary Row Operations
We can perform the following operations on the matrix
3. Adding a multiple of one row to another. 1 1 1 44
52 2 5 11 2 2 11
4 6 8 24 6 84 24
x y z
x y z
x y z
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.
1 2R R
2 1 22R R R
1 12R R
Gauss-Jordan Elimination
Main Idea: We want to use elementary row operations to get the matrix into the form
1 0 0 *
0 1 0 *
0 0 1 *
Gauss-Jordan Elimination
Main Idea: We want to use elementary row operations to get the matrix into the form
1 0 0 *
0 1 0 *
0 0 1 *
1 2
3
Example 2
Use Gauss-Jordan Elimination to solve the system.
1 1 1 4
2 2 5 11
4 6 8 24
Example 2
1 1 1 4
52 2 11
6 84 24
x
y
z
1 0 0 *
0 1 0 *
0 0 1 *
1 2
3
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!!
Example 3
Use Gauss-Jordan Elimination to solve the system.
3 3 1
2 3
2 3 4
x y z
x y z
x y z
Example 3
x
y
z
3 31 1
31 2 1
32 1 4
1 0 0 *
0 1 0 *
0 0 1 *
1 2
3
How do I Confirm My Answer?
