1. Chapter 4
Matrices

2. 4.1 Intro to Matrices
Matrix: a rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets
Element: a value in a matrix
Dimensions: number of rows x number of columns
Read “m by n”

3. State the dimensions of matrix G if
State the dimensions of matrix A if
A =

4. Types of Matrices Row matrix: Column matrix: Square matrix:
A matrix with only one row ex:
Column matrix:
A matrix with only one column ex:
Square matrix:
A matrix with the same number of rows as columns
ex: −4
Zero matrix:
All elements are zero

5. Solve an equation involving Matrices
1. 𝑦 3𝑥 = 6−2𝑥 31+4𝑦

6. 𝑥+2 𝑦−4 0 4𝑧+6 = 12 −8 0 2

7. NOW PLAYING
Ticket Information
Evening Shows
Matinee Shows
Adult …. $7.50
Adult …. $5.50
Child ….$4.50
Senior….$5.50
Senior ….$5.50
Twilight Shows
All tickets…….$3.75 3.
Write a matrix for the prices of movie tickets for adults, children, and seniors.
What are the dimensions of the matrix?

8. 4.2 Operations with matrices

9. Matrices can be added and subtracted if, and only if, they have the same dimensions.
ex: 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ = 𝑎+𝑒 𝑏+𝑓 𝑐+𝑔 𝑑+ℎ
ex: 𝑎 𝑏 𝑐 𝑑 − 𝑒 𝑓 𝑔 ℎ = 𝑎−𝑒 𝑏−𝑓 𝑐−𝑔 𝑑−ℎ

10. 1. Find A + B if A = − and B = −

11. 2. Find A + B if A = −6 7 −9 3 and B = 4 −2 0 1 5 −1

12. 3. Find B – A if A = and B = −3 0

13. Scalar Multiplication
Scalar: a constant that you can multiply a matrix by
ex: x 𝑎 𝑏 𝑐 𝑑 = 𝑥𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑑

14. 4. Find 3A, if A = − −

15. 5. If A = − and B = 2 −3 5 −4 , find 5A – 2B

16. 4.3 Multiplying matrices

17. Ex: A5 x 3 and B3 x 4 = AB You can multiply matrices if and only if:
the number of columns in the first matrix is the same as the number of rows in the second matrix
Ex: A5 x 3 and B3 x 4 = AB
If the matrices cannot be multiplied =
product matrix is not defined
5 x 4

18. Multiplying Matrices 𝑎 𝑏 𝑐 𝑑 x 𝑤 𝑥 𝑦 𝑧 = 𝑎𝑤+𝑏𝑦 𝑎𝑥+𝑏𝑧 𝑐𝑤+𝑑𝑦 𝑐𝑥+𝑑𝑧
Step 1: 𝑎 𝑏 𝑐 𝑑 x 𝑤 𝑥 𝑦 𝑧 Step 2 : 𝑎 𝑏 𝑐 𝑑 x 𝑤 𝑥 𝑦 𝑧 Step 3 : 𝑎 𝑏 𝑐 𝑑 x 𝑤 𝑥 𝑦 𝑧 Step 4 : 𝑎 𝑏 𝑐 𝑑 x 𝑤 𝑥 𝑦 𝑧

19. Find RS if
1. R = − and S = −

20. At a swimming meet 6 points are awarded for 1st place, 4 points for 2nd place, and 3 points for 3rd place.
2. The chart shows how many swimmers placed in each position through the meet for the four participating schools. Write a set of matrices to model the points earned. Which team won the meet?
School
1st Place
2nd Place
3rd Place
Central Dauphin 4 7 3
Cumberland Valley 8 1
Hershey 10 5
Carlisle 6

21. Commutative Property – Does it work for matrices?
3. Find each product if P = −1 and S = 9 −3 2 6 −1 −5 a. PS b. SP

22. Distributive Property – Does it work for matrices?
4. Find each product if A = 3 2 −1 4 B = − and C = 1 1 −5 3 a. A (B + C) b. AB + AC

23. 4.5 Determinants

24. Determinant: A number associated with a square matrix
Second-Order Determinant
A value found by calculating the difference of the products of the two diagonals in a 2x2 matrix
𝑎 𝑏 𝑐 𝑑 = ad – bc

25. Find the value of the determinant
1. − −3 2

26. −

27. Third-Order Determinant
Determinant of a 3x3 matrix
Method 1: Expansion by Minors
𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 = a 𝑒 𝑓 ℎ 𝑖 - b 𝑑 𝑓 𝑔 𝑖 + c 𝑑 𝑒 𝑔 ℎ
Method 2: Diagonals

28. Find the determinant using expansion by minors
−3 −1 5 −

29. 4.6 Cramer’s rule

30. Use the determinants to solve systems of equations
Ex: ax + by = e
cx + dy = f
x = 𝑒 𝑏 𝑓 𝑑 𝑎 𝑏 𝑐 𝑑 and y = 𝑎 𝑒 𝑐 𝑓 𝑎 𝑏 𝑐 𝑑 Write the answer as (x, y)
x = 𝑑𝑒 −𝑏𝑓 𝑎𝑑 −𝑏𝑐 and y = 𝑎𝑓 −𝑐𝑒 𝑎𝑑 −𝑏𝑐

31. Solve the system of equations using Cramer’s rule
1. 5x + 4y = 28 3x – 2y = 8

32. 2. 2x – 3y = 12 -6x + y = -20

33. In voting for the colors of a new high school, blue & gold received 440 votes from 10th and 11th graders while red & black received 210 votes from the same grades. In the 10th grade, blue & gold received 72% of the total and Red & black received 28%. In the 11th grade, Blue & gold received 64% of the total and Red & Black received 36%.
Write a system of equations that represents the total number of votes for each pair of colors.
Find the total number of votes cast in 10th grade and in 11th grade.

34. 4.7 Identity and Inverse Matrices

35. Identity Matrix:
A square matrix that, when multiplied by another matrix equals the same matrix
Ex: or

36. Inverse Matrices:
When the product of two matrices with the same dimensions is the identity matrix

37. Determine whether each pair of matrices are inverses of each other.
1. X = − and Y = −1 1 4

38. 2. C = and D = 1 −2 −

39. To find the inverse of a matrix 𝑎 𝑏 𝑐 𝑑
Find the determinant to see if it has an inverse 𝑎𝑑 −𝑏𝑐
If the determinant is zero, it cannot have an inverse
If the inverse exists it = 𝑎𝑑−𝑏𝑐 𝑑 −𝑏 −𝑐 𝑎

40. Find the inverse for the given matrix

41. 4. A = −4 3

42. 4.8 Using Matrices to solve systems of equations

43. Step 1: Rewrite the system of equations as a matrix equation
Ex: 5x + 7y = 11
3x + 8y = 18
∙ 𝑥 𝑦 =
Step 2: find the inverse matrix
1 −40 − −7 − = 1 − −7 −3 5 = − −5 61

44. Step 4: Write the solution as an ordered pair : 38 61 , −57 61
Step 3: Multiply each side of the matrix equation by the inverse matrix
− − ∙ ∙ 𝑥 𝑦 = − − ∙
∙ 𝑥 𝑦 = − (18) −5 61 (18)
𝑥 𝑦 = −57 61
Step 4: Write the solution as an ordered pair : , −57 61

45. Solve the system using matrices
1. 5x + 3y = 13 4x + 7y = -8

46. 2. 6a – 9b = -18 8a – 12b = 24