10/11/2014 5:29 AM4.5 - Matrix Inverse and Solving Systems1 Warm-Up #2 Find the determinants: 1. 2....

04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 1

Warm-Up #2

Find the determinants:1.

2.

Use Cramer’s Rule to determine the solution:

3.

6 –23 4


DBA 10th graders

1 D

2 H

3 C

4 G

5 C

6 J

7 A

8 G

9 B

10 H

11 B

12 G

13 C

14 G

15 B

16 G

17 C

18 G

19 D

20 G

21 C

22 F

23 D

24 G

25 A


DBA 11th graders

1 D

2 H

3 C

4 H

5 B

6 J

7 D

8 G

9 B

10 H

11 B

12 G

13 D

14 H

15 B

16 H

17 B

18 G

19 C

20 F

21 B

22 H

23 C

24 F

25 C


Pg 274

6. (1/5, -4/5)

7. No solution

8. Omit

9. (5/2, -1/4)

18. No solution

19. (1.27, 1.11)

20. No solution

21. (-14/3, 7/9)

25. Omit

26. 63.75 sq. miles

27. 17 sq units


Matrix Inverses and Solving Systems

Section 4.5 - [email protected] ©2007


Definition• 1 is the multiplicative inverse for real

numbers because multiplying by a real number will make it equal to 1

• Example: 1/3 • x = 1 x = 3


Inverse Matrices

Inverse of the matrix is…

• Multiply the with the

inverse matrix

a bA

c d

1

a b

c d ad bc

1

ad bc


Example 1

1

a b

c d ad bc

12 7 1

5 3 (12)(3) (7)(5)

Find the inverse of: 12 7

5 3


Example 1


5 3

12 7 3 –71

5 3 –5 1236 35

12 7 3 –7(1)

5 3 –5 12

3 –7

–5 12


Example 2


1 3

3 –221

– 12

Determine whether these two matrices are inverses:


Example 3

-2 2

3 4

2 1

3 / 2 1

and

-2 2

3 4

1

( )ad bc1

2

4 2

3 2

=2 1

3 / 2 1

yes

Determine whether these two matrices are inverses:


Your Turn

1 5

4 8

1/12 5/12

1/ 2 2 / 3

and

no


Solving Matrix Equation

When solving for variables in a matrix:1. Write a matrix equation with the

coefficients in one bracket and variables on one side and constants on other side

2. Find the inverse3. Multiply the inverse with the

constant and add4. Check

Write a matrix equation for the system and solve using inverse matrix


Example 4

3 15

3 2 6

x y

x y

3 1

3 2

Step 1: Write the equation in matrix form

x

y

15

6

=



Example 4

3 15

3 2 6

x y

x y

3 1

3 2

Step 2: Find the inverse

= -2 11

–3 3(3)( 2) (1)(3)

2 11

–3 39

2 / 9 1/ 9

1/3 1/3



Example 4

3 15

3 2 6

x y

x y

Step 3: Multiply the inverse with the constants and add them

15

6

(10 / 3) (2 / 3)

4

2 / 9 1/ 9

1/3 1/3

(10 / 3) (2 / 3)

5 + ( 2)

4

3

(4, 3)



Example 4

3 15

3 2 6

x y

x y

Step 4: Check

3(4) (3) 12 3 15 15

3(4) 2(3) 12 6 6 6



Example 5

Step 1: Write the equation in matrix form



Example 5

Step 2: Find the inverse



Example 5

Step 3: Multiply the inverse with constants

(5, –2)



Your Turn

(3, 1)

April paid $24 for 2 admission tickets to see a concert and 7 soft drinks. Alexander paid $46 for 4 admission tickets and 13 soft drinks. Use an inverse matrix to determine the amount of money of each admission ticket and soft drink.


Example 6

x = tickets, y = drinksIdentify the variables

Setup the system2 7 24

4 13 46

x y

x y

Solve for x 2 7 24

4 13 46

x

y

1

2

13 –71;

–4 226 28D

April paid $24 for 2 admission tickets to see a concert and 7 soft drinks. Alexander paid $46 for 4 admission tickets and 13 soft drinks. Use an inverse matrix to determine the amount of money of each admission ticket and soft drink.


Example 6

Solve for x

13 724–

2 246

2 –1

5

2

x

y

1

2

13 –71;

–4 226 28D

Each ticket was worth $5 and each soft drink was worth $2

Pg 282 15, 17-21 odd, 22-24 all, 27a-d


Assignment

10/11/2014 5:29 AM4.5 - Matrix Inverse and Solving Systems1 Warm-Up #2 Find the determinants: 1. 2....

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