matrix properties and equations activity more matrix math 1 · 2019. 9. 18. · unit 1 • linear...

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ACTIVITY

Unit 1 • Linear Systems and Matrices 57

1.7Matrix Properties and EquationsMore Matrix MathSUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge

Th ere are many properties that hold true for real numbers. Some of these same properties also hold true for matrices.

1. Use these matrices.

A = ⎡

⎢

⎣

4 5

2 -2

⎤

�

⎦

B = ⎡

⎢

⎣

5 2

6 4

⎤

�

⎦

a. Find A + B and B + A.

b. Find AB and BA.

2. For any two real numbers a and b, it is true that a + b = b + a andab = ba.

a. What property do these statements demonstrate?

b. Does this property hold true for matrix addition and matrix multiplication? Explain.

57-66_SB_A2_1-7_SE_Act.indd 5757-66_SB_A2_1-7_SE_Act.indd 57 2/2/10 10:20:50 AM2/2/10 10:20:50 AM

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58 SpringBoard® Mathematics with Meaning™ Algebra 2

My Notes

Matrix Properties and Equations ACTIVITY 1.7continued More Matrix MathMore Matrix Math

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge

3. Decide whether matrix addition and multiplication are associative. Justify your answers.

4. Suppose that A is any m × n matrix.

a. Th e identity matrix for addition is a matrix O such that A + O = A and O + A = A. What is the dimension and what are the elements of matrix O?

b. Th e additive inverse for matrix A is the matrix -A such that A + (-A) = O. Describe -A.

Recall the following properties for real numbers a, b, and c.

Addition Properties

Commutativea + b = b + a

Associative(a + b) + c = a + (b + c)

Identitya + 0 = a, 0 + a = a

Inversea + (-a) = 0

Multiplication Properties

Commutativeab = ba

Associative(ab)c = a(bc)

Identitya · 1 = a, 1 · a = a

Inverse

a · 1 __ a = 1 , a ≠ 0


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SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge

ACTIVITY 1.7continued

Matrix Properties and Equations More Matrix MathMore Matrix Math

5. Use these matrices.

B = ⎡

⎢

⎣

5 2

6 4

⎤

�

⎦

C = ⎡

⎢

⎣

1 0

0 1

⎤

�

⎦

D = E = ⎡

⎢

⎣

1

0 0

0

1 0

0

0 1

⎤

�

⎦

a. Find BC and CB.

b. Find DE and ED.

c. For a square matrix, describe what the multiplicative identity matrix I appears to be.

⎡

⎢

⎣

5

4 2

0

1 2

-2

3 1

⎤

�

⎦


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SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer

Each square matrix is associated with a real number called the

determinant. For a 2 × 2 matrix ⎡

⎢

⎣

a c b

d ⎤

�

⎦

the determinant is ad - bc.

6. Use the defi nition to fi nd the determinant of ⎡

⎢

⎣

4 5

-2 -2

⎤

�

⎦

.

Every square matrix A has a multiplicative inverse A-1, except when det A = 0.

For the matrix A = ⎡

⎢

⎣

a c b

d ⎤

�

⎦

, if det A ≠ 0,

then A-1 = 1 _____ det A ⎡

⎢

⎣

d -c -b a

⎤

�

⎦

= 1 _______ ad - bc ⎡

⎢

⎣

d -c -b a

⎤

�

⎦

.

7. For A = ⎡

⎢

⎣

4 5

-2 -2

⎤

�

⎦

, use the defi nition above to fi nd A-1.

8. Explain in words how to fi nd the inverse of a 2 × 2 matrix.

In the set of real numbers, every real number has an additive inverse -a and every real num-ber except 0 has a multiplicative

inverse 1 __ a .

WRITING MATH

You can write two different symbols for the determinant.

| a

c

b

d | det A

ACADEMIC VOCABULARY

determinant


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9. Find A · A-1 and A-1 · A. Explain why these products verify that the matrices are inverses.

TRY THESE A

Find the determinant and inverse of each matrix, if possible. If not possible, explain why. Write your answers in the My Notes space. Show your work.

a. A = ⎡

⎢

⎣

-2 -5

3 8

⎤

�

⎦

b. B = ⎡

⎢

⎣

8 4

6 2

⎤

�

⎦

c. C = ⎡

⎢

⎣

-4 1

-8 2

⎤

�

⎦

Th e County Feed Store stocks two diff erent types of feed, which they use to mix for specifi c customer orders. Th e percent of fi ber and protein for the two types of feed is shown in the table below.

Percent Fiber Percent ProteinType A 40 30Type B 15 45

A customer orders 3000 lb of mixture with 28% fi ber.

10. Let a represent the number of pounds of Type A feed used in the mixture and b represent the number of pounds of Type B feed used in the mixture.

a. Write a system of equations that can be used to determine how much of each feed type is needed to fi ll the customer’s order.

SUGGESTED LEARNING STRATEGIES: Quickwrite, Marking the Text, Create Representations


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10. (continued)

b. Use an algebraic method to solve the system of equations and fi nd the amount of each type of feed.

Matrices can be used to represent systems of equations. Th e matrices in the equation below are called the coeffi cient matrix, the variable matrix, and the constant matrix, respectively.

⎡

⎢

⎣

______ ______

______ ______ ⎤

�

⎦

· ⎡

⎢

⎣

a b

⎤

�

⎦

= ⎡

⎢

⎣

______ ______ ⎤

�

⎦

11. Complete the matrix equation above to represent the system of equations in Item 10.

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Summarize/Paraphrase/Retell, Create Representations


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SUGGESTED LEARNING STRATEGIES: Think/Pair/Share

To solve the matrix equation A · X = B, multiply both sides of the matrix equation by A-1, where the matrix A-1 is the multiplicative inverse of A. Th e solution of the matrix equation is X = A-1 · B.

12. Use the matrix equation from Item 11.

a. What are matrices A, X, and B?

b. Find A-1, the inverse of matrix A.

c. Use matrix multiplication to determine A-1 · B

To solve an equation of the type a · x = b for x, where a, b, and x are real numbers and a ≠ 0, multiply each side of the equa-tion by 1 __ a , also known as a-1, the multiplicative inverse of a. This method produces a solution that can be written as x = a-1 · b.

For example:

4x = 20 ax = b

1 __ 4 · 4x = 1 __ 4 · 20

1 · x = 1 __ 4 · 20

x = 1 __ 4 · 20 x = a-1 · b

x = 5


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12. (continued)

d. How does the matrix found in Part (c) compare to the answers found in Item 10(b)?

13. Use a graphing calculator and the given matrices.

A = ⎡

⎢

⎣

4 5

-2 -2

⎤

�

⎦

F =

⎡

⎢

⎣

1

3

-5

4

7 6

-2

10 8

⎤

�

⎦

a. Find det A and A-1. Verify your answers to Items 6 and 7.

b. Find det F and F-1.

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share

Finding the determinant and inverse of a square matrix with dimension larger than 2 × 2 is time-consuming. A graphing calculator makes the task easier.

TECHNOLOGY


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SUGGESTED LEARNING STRATEGIES: Create Representations, Marking the Text

14. Write the system of equations represented by the matrix equation below. Th en solve by using a graphing calculator.

⎡

⎢ ⎣

1

3

-5

4

7 6

-2

10 -8

⎤

�

⎦

⎡

⎢

⎣

x

y z ⎤

�

⎦

= ⎡

⎢

⎣

5

21 9

⎤

�

⎦

Th e County Feed Store has added a third type of feed.

Percent Fiber Percent ProteinType A 40 30Type B 15 45Type C 25 15

15. For each customer order, let a represent the number of pounds of Type A feed used in the mixture, let b represent the number of pounds of Type B feed used in the mixture, and let c represent the number of pounds of Type C feed used in the mixture.

a. Write a system of three equations for an order that consists of 1200 lb of feed with 25% fi ber and 20% protein.


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15. (continued)

b. Defi ne matrices A, X, and B so that the three equations from Part (a) can be written as a matrix equation of the form A · X = B.

c. Use a graphing calculator to solve the matrix equation for X. Identify the amount of each type of feed used in the mixture.

SUGGESTED LEARNING STRATEGIES: Create Representations

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. Let A and B be two matrices with

A + B = ⎡

⎢

⎣

2 3

9 5

⎤

�

⎦

. If possible, determine the

value of B + A. If not possible, explain why.

2. Let A and B be two matrices with

A · B = ⎡

⎢

⎣

-3 9

9 15

⎤

�

⎦

. If possible, determine the

value of B · A. If not possible, explain why.

3. Let A = ⎡

⎢

⎣

2 3

5 7

⎤

�

⎦

and B = ⎡

⎢

⎣

-7 3

5 -2

⎤

�

⎦

. Find AB

and BA. Th en explain what these products tell you about matrices A and B.

4. Determine whether the inverse of each matrix exists. If an inverse exists, fi nd it.

a. ⎡

⎢ ⎣ 1 2

3 5

⎤

�

⎦

b.

⎡

⎢ ⎣

7 2

14 4

⎤

�

⎦

c. ⎡

⎢ ⎣

3 -4

2 -2

⎤

�

⎦

5. Write the system of equations represented by the matrix equation below. Th en solve the matrix equation.

⎡

⎢

⎣

4 1

-2 2

⎤

�

⎦

⎡

⎢

⎣

x y

⎤

�

⎦

= ⎡

⎢

⎣

7 3

⎤

�

⎦

6. Th e table shows dietary data for 1 oz of nuts.

Nut Fat Carbohydrates CaloriesAlmonds 7 g 12 g 105Cashews 14 g 10 g 165Walnuts 18 g 5 g 185

Set up a matrix equation to fi nd the number of ounces of each type of nut that can be combined to produce a mixture with 68 g fat, 60 g carbohydrates, and 835 calories. Use a graphing calculator to solve the equation and fi nd the amount of each type of nut in the mixture to the nearest tenth of an ounce.

7. MATHEMATICAL R E F L E C T I O N

Describe which properties of real numbers hold for

matrices.


matrix properties and equations activity more matrix math 1 · 2019. 9. 18. · unit 1 • linear...

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