Lesson 4-1 Introduction to Matrices

Lesson 4-2 Operations with Matrices

Lesson 4-3 Multiplying Matrices

Lesson 4-4 Transformations with Matrices

Lesson 4-5 Determinants

Lesson 4-6 Cramer's Rule

Lesson 4-7 Identity and Inverse Matrices

Lesson 4-8 Using Matrices to Solve Systems of Equations

Five-Minute Check (over Chapter 3)

Main Ideas and Vocabulary

Example 1: Real-World Example: Organize Data intoa Matrix

Example 2: Dimensions of a Matrix

Example 3: Solve an Equation Involving Matrices

• matrix

• element

• dimension

• row matrix

• column matrix

• square matrix

• zero matrix

• Organize data in matrices.

• Solve equations involving matrices.

• equal matrices

COLLEGE Kaitlin wants to attend one of three Iowauniversities next year. She has gathered information about tuition (T), room and board (R/B), and enrollment (E) for the universities. Use a matrix to organize the information. Which university’s total cost is lowest?Iowa State University:

T - $5426 R/B - $5958 E - 26,380

University of Iowa:

T - $5612 R/B - $6560 E - 28,442

University of Northern Iowa:

T - $5387 R/B - $5261 E - 12,927

Organize Data into a Matrix

Organize the data into labeled columns and rows.

Answer: The University of Northern Iowa has the lowest total cost.

Organize Data into a Matrix

ISU

UI

UNI

T R/B E

DINING OUT Justin is going out for lunch. The information he has gathered from two fast-food restaurants is listed below. Use a matrix to organize the information. When is each restaurant’s total cost less expensive?

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. The Burger Complex has the best price for chicken sandwiches. The Lunch Express has the best prices for hamburgers and cheeseburgers.

B. The Burger Complex has the best price for hamburgers and cheeseburgers. The Lunch Express has the best price for chicken sandwiches.

C. The Burger Complex has the best price for chicken sandwiches and hamburgers. The Lunch Express has the best prices for cheeseburgers.

D. The Burger Complex has the best price for cheeseburgers. The Lunch Express has the best price for chicken sandwiches and hamburgers.

Dimensions of a Matrix

Answer: Since matrix G has 2 rows and 4 columns, the dimensions of matrix G are 2 × 4.

4 columns

2 rows

State the dimensions of matrix G if

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. 2 × 3

B. 2 × 2

C. 3 × 2

D. 3 × 3

State the dimensions of matrix G if G =

Solve an Equation Involving Matrices

Since the matrices are equal, the corresponding elements are equal. When you write the sentences to solve this equation, two linear equations are formed.

y = 3x – 2

3 = 2y + x

Solve an Equation Involving Matrices

This system can be solved using substitution.

3 = 2y + x Second equation

3 = 2(3x – 2) + x Substitute 3x – 2 for y.

3 = 6x – 4 + x Distributive Property

7 = 7x Add 4 to each side.

1 = x Divide each side by 7.

Solve an Equation Involving Matrices

To find the value for y, substitute 1 for x in either equation.

y = 3x – 2 First equation

y = 3(1) – 2 Substitute 1 for x.

y = 1 Simplify.

Answer: The solution is (1, 1).

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. (2, 5)

B. (5, 2)

C. (2, 2)

D. (5, 5)

http://www.algebra2.com/

Five-Minute Check (over Lesson 4-1)

Main Ideas and Vocabulary

Key Concept: Addition and Subtraction of Matrices

Example 1: Add Matrices

Example 2: Subtract Matrices

Example 3: Real-World Example

Key Concept: Scalar Multiplication

Example 4: Multiply a Matrix by a Scalar

Concept Summary: Properties of Matrix Operations

Example 5: Combination of Matrix Operations

• scalar

• scalar multiplication

• Add and subtract matrices.

• Multiply by a matrix scalar.

http://www.algebra2.com/

Add Matrices

Definition of matrix addition

Add corresponding elements.

Simplify.

Answer:

Add Matrices

Answer: Since the dimensions of A are 2 × 3 and the dimensions of B are 2 × 2, these matrices cannot be added.

A. A

B. B

C. C

D. D

A B C D

0% 0%0%0%

A.

B.

C.

D.

A. A

B. B

C. C

D. D

A B C D

0% 0%0%0%

A.

B.

C.

D.

Subtract Matrices

Definition of matrix subtraction

Subtract corresponding elements.

Simplify.

Answer:

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A.

B.

C.

D.

SCHOOL ATHLETES The table below shows the total number of student athletes and the number of female athletes in three high schools. Use matrices to find the number of male athletes in each school.

The data in the table can be organized into two matrices. Find the difference of the matrix that represents the total number of athletes and the matrix that represents the number of female athletes.

Subtract corresponding elements.

total female male

Answer: There are 582 male athletes at Jefferson, 286 male athletes at South, and 677 male athletes at Ferguson.

TESTS The table below shows the percentage of students at Clark High School who passed the 9th and 10th grade proficiency tests in 2001 and 2002. Use matrices to find how the percentage of passing students changed from 2001 to 2002.

A B C D

0% 0%0%0%

1. A

2. B

3. C

4. D

A. B.

C. D.

9th grade 10th grade

Math

Reading

Science

Citizenship

Math

Reading

Science

Citizenship

Math

Reading

Science

Citizenship

Math

Reading

Science

Citizenship

9th grade 10th grade

9th grade 10th grade 9th grade 10th grade

http://www.algebra2.com/

Substitution

Multiply a Matrix by a Scalar

Multiply each element by 2.

Multiply a Matrix by a Scalar

Answer:

Simplify.

A. A

B. B

C. C

D. D

A B C D

0% 0%0%0%

A. B.

C. D.

http://www.algebra2.com/

Perform the scalar multiplication first. Then subtract the matrices.

Combination of Matrix Operations

Substitution

Multiply each element in the first matrix by 4 and each element in the second matrix by 3.

4A – 3B

Combination of Matrix Operations

Simplify.

Subtract corresponding elements.

Answer:

Simplify.

A. A

B. B

C. C

D. D

A B C D

0% 0%0%0%

A.

B.

C.

D.