Post on 27-Mar-2015
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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)
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.
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.
Subtract Matrices
Definition of matrix subtraction
Subtract corresponding elements.
Simplify.
Answer:
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
Multiply each element by 2.
Multiply a Matrix by a Scalar
Answer:
Simplify.
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.