Meiling chensignals & systems1 Lecture #2 Introduction to Systems.
10/11/2014 5:29 AM4.5 - Matrix Inverse and Solving Systems1 Warm-Up #2 Find the determinants: 1. 2....
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Transcript of 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
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 2
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
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 3
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
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 4
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
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 5
Matrix Inverses and Solving Systems
Section 4.5 - [email protected] ©2007
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 6
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
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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
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 8
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
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 9
Example 1
Find the inverse of: 12 7
5 3
12 7 3 –71
5 3 –5 1236 35
12 7 3 –7(1)
5 3 –5 12
3 –7
–5 12
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 10
Example 2
Find the inverse of: 2 4
1 3
3 –221
– 12
Determine whether these two matrices are inverses:
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 11
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:
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 12
Your Turn
1 5
4 8
1/12 5/12
1/ 2 2 / 3
and
no
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 13
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
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 14
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
=
Write a matrix equation for the system and solve using inverse matrix
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 15
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
Write a matrix equation for the system and solve using inverse matrix
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 16
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)
Write a matrix equation for the system and solve using inverse matrix
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 17
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
Write a matrix equation for the system and solve using inverse matrix
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 18
Example 5
Step 1: Write the equation in matrix form
Write a matrix equation for the system and solve using inverse matrix
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 19
Example 5
Step 2: Find the inverse
Write a matrix equation for the system and solve using inverse matrix
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 20
Example 5
Step 3: Multiply the inverse with constants
(5, –2)
Write a matrix equation for the system and solve using inverse matrix
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 21
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.
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 22
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.
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 23
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
04/11/23 01:23 4.5 - Matrix Inverse and Solving Systems 24
Assignment