Lesson 2.3: Modeling Data with Matrices – day 1
SAT Practice
Grid-In: How would you grid the following answers?
3
1. 15
2. 2
3. 2.75
4
4.
0.33333
5.
14
22
Review
Classify the following systems:
y  2x  7
1. 
4x  2y  6
6.
9 < y < 9.1
2.
 y  2x  3

4x  2y  6
Essential Question
How do you represent data in a matrix and how do you find
sums, differences, and scalar multiples of matrices?
Definitions
 Matrix: A rectangular array of numbers enclosed in a single
set of brackets.
 Dimensions: The number of horizontal rows and the number
of vertical columns. Always written as rows x columns.
 Element: Each individual number in the matrix, also called
an entry.
Special Names are given to certain matrices
 Row Matrix: a matrix with only one row
 Column Matrix: a matrix with only one column
 Square Matrix: a matrix with the same numbers of rows
and columns
Example 1:
Matrix S represents the sales at a music store.
a) What are the dimensions of
Record Tape CD
Matrix S?
Rock
1
15 30
b)
c)
What does element S 4,2
refer to?
What does element S 5,1
refer to?
Jazz
10
10
5
Blues
15
3
4
Country
2
20
25
Classical
1
3
1
=S
You Try: During the summer, Mrs. Robbins received several
types of grains on her farm to feed her livestock. June 15,000 bushels corn, 2,000 bushels soybeans, 500 bushels oats;
July – 13,500 bushels corn, 6,500 bushels soybeans, 1,000
bushels oats; August – 14,000 bushels corn, 5,500 bushels
soybeans, 1,500 bushels oats.
a. Use a matrix to represent the data.
b. Use a symbol to represent the number of bushels of
soybeans in August.
 Equivalent Matrices: Two matrices are equal if they have
the exact same dimensions and all their corresponding
elements are equal.
Example 2: Solve for x and y.
4x+5
9
7 -2y+3
You Try:
y
  4x 
 y  3  2x  1

 

15
-1
=
21
7
9
y-12
15
-1
 Adding and Subtracting: Matrices can be added and
subtracted if and only if they have the exact same
dimensions. To find the sum (or difference) you add (or
subtract) the corresponding entries of the matrices.
Properties of Matrix Addition
Let A, B, and C, each be matrices with dimensions m x n. Z is a
matrix with dimensions m x n where all the entries are 0.
Commutative:
A+B=B+A
Associative:
A + (B + C) = (A + B) + C
Identity:
A+Z=A
Inverse:
A + (-A) = Z
 Scalar Multiplication: When the same number multiplies
each entry in a matrix.
Example 3:
M=
-5
11
9
-1
2
3
N=
2
22
a. Find M + N
b. Find M – N
c. Find M – M
d. Find 2M
6
10
-31
0
You Try:
 7 4 


A 5 0 
 3 1 


a. Find A + B
 6 10 


9 
B  8
 2 5 


b. Find A – B
c. Find B – A
d. Find 2B
Point Matrices: A matrix can be used to represent a figure
graphed in a coordinate plane. It will have the dimensions of 2
horizontal rows (one for x-coordinate and one for the ycoordinate) by the number of points that are in the figure (this
will be the amount of vertical columns.) Once again, it will be a
2 x n matrix, NOT n x 2!
Example 4. Give the matrix, P, for the following quadrilateral.
b. Graph ½P on the graph.
Lesson 2.3: Modeling Data with Matrices – day 2
SAT Practice
Grid-In: What is the value of x?
xo
2xo
Review
A clothing store took inventory of its shirts and found the
following:
Jerseys: 12 small, 28 medium, and 17 large
T-shirts: 15 small, 32 medium, and 45 large
Sweatshirts : 6 small, 20 medium, and 30 large
a) Arrange the data in a matrix and call it M.
b) What are the dimensions of matrix M?
c) Interpret the entry at M2,3.
d) Calculate 3M.
Essential Question
How do you multiply matrices?
Matrix Multiplication
 inner dimensions must be the same
 answer will have the outer dimensions
a b  e f g  ae  bh af  bi
c d  h i j   ce  dh cf  di


 
2x2
2x3
2x3
ag  bj 
cg  dj 
match!
answer’s dim.
Examples
2
3   4
1
You Try:  1  2  3   5 

 
 4
5
6    6
1.
3 9  7 0
2  1  1 3



2.
Give an example of two matrices that cannot be multiplied.
3.
Is matrix multiplication commutative? Justify your answer.
Ex 4.
 a 5
 2 b   6

 d
 c 0 
1 
 4
Ex 5. At the Ohio State University, professional students pay
different tuition rates based on the programs they have
chosen. For the 2002-2003 school year, Medical students paid
$5,456 per quarter, dental students paid $4,792 per quarter,
and veterinary students paid $4,405. The chart lists the total
enrollment in those programs. Use matrix multiplication to find
the amount of tuition paid for each of these four quarters.
Enrollment
Quarter Med. Dent. Vet.
Autumn 826
400 537
Winter
818
401 537
Spring
820
399 536
Summer 425
205 135