1

Systems of Linear Equations & Matrices Sections

4.2 & 4.3

After today’s lesson, you will be able to Use terms associated with matrices. Set up and solve the augmented matrix associated with a

linear system in two variables on a calculator. Identify the three possible matrix solution types for a linear

system in two variables. Convert a matrix to reduced row echelon form using the

calculator.

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Matrix Methods

It is impractical to solve more complicated linear systems by hand. Computers and calculators now have built in routines to solve larger and more complex systems.

Matrices, in conjunction with graphing utilities and or computers are used for solving more complex systems.

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Matrices

A matrix is a rectangular array of numbers written within brackets.

The subscripts give the “address” of each entry of the matrix. For example the entry a23 is found in the second row and third column

11 12 13 14

21 22 23 24

a a a a

a a a a

Each number in the matrix is called an element.

Since this matrix has 2 rows and 4 columns, the dimensions of the matrix are 2 x 4.

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Matrix Solution of Linear Systems

We can represent a linear system of equations by an augmented matrix, a matrix which stores the coefficients and constants of the linear system and then manipulate the augmented matrix to obtain the solution of the system.

Example: Write the augmented matrix for the system:

3 5 9

6 1 2

The augmented matrix associated with the above system is:

Make sure the system’s equations are in general form before creating a matrix.

3 5 9

6 2

x y

x y

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Reduced Row Echelon Form

An RREF matrix has 1’s down its main diagonal and 0’s below and above the 1’s.

To solve the system of linear equations, we want to convert our augmented matrix to a matrix in reduced row-echelon form. We will use our graphing calculators to do this.

The reduced row-echelon form of the previous matrix is:

11 0

3320

0 111

See handout for directions on how to obtain the rref matrix using your calculator.

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Writing a Solution from a Matrix in Reduced Row Echelon Form

1 0 0 13.21 0 3

1) 2) 0 1 0 15.60 1 17

0 0 1 8.7

Example: Write the solution to the system represented by the each of the following matrices.

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Using Subscripted Variables

Sometimes, instead of using different letters to represent the unknown quantities, we use the same variable with different subscripts.

Example: Solve the system using matrices. Round to hundredths.

1 2

1 2

5.70 8.55 35.91

4.50 5.73 76.17

x x

x x

a) Write the augmented matrix for the system.

b) Using your calculator, find the RREF matrix for the system.

c) Write your conclusion about the system.

8

Continued

1 2

1 2

5.70 8.55 35.91

4.50 5.73 76.17

x x

x x

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Example

2 2

2 2 5

2 7

x y z

x y z

z y x

Example: Given the following system, complete the following parts.

a) Write the augmented matrix for the system

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Example

b) Using your calculator, find the RREF matrix for the system

c) Write your conclusion about the system.

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Example

Example: Given the following system, complete the following parts.

a) Write the augmented matrix for the system

b) Using your calculator, find the RREF matrix for the system

c) Write your conclusion about the system.

10 2 6

5 3

x y

y x

12

Continued

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Example

Example: Given the following system, complete the following parts.

a) Write the augmented matrix for the system

b) Using your calculator, find the RREF matrix for the system

c) Write your conclusion about the system.

3

2 1

5 1

x y

y z

x z

14

Continued

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Possible Final Matrix Forms for a Linear System in Two Variables

Form 1: Unique Solution (Consistent and Independent)

1 0

0 1

a

b

Form 2: Infinitely Many Solutions (Consistent and Dependent)

1

0 0 0

a b

Form 3: No Solution (Inconsistent and Independent)

1

0 0

a b

c