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Holt Algebra 2
4-1 Matrices and Data 4-1 Matrices and Data
Holt Algebra 2
Warm Up
Lesson Presentation
Lesson Quiz
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Holt Algebra 2
4-1 Matrices and Data
The table shows the top scores for girls in barrel racing at the 2004 National High School Rodeo finals. The data can be presented in a table or a spreadsheet as rows and columns of numbers.
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Holt Algebra 2
4-1 Matrices and Data
You can also use a matrix to show table data. A matrix is a rectangular array of numbers enclosed in brackets.
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Holt Algebra 2
4-1 Matrices and Data
Use matrices to display mathematical and real-world data.
Find sums, differences, scalar products and products of matrices.
Objectives
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Holt Algebra 2
4-1 Matrices and Data
![Page 6: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/6.jpg)
Holt Algebra 2
4-1 Matrices and Data
![Page 7: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/7.jpg)
Holt Algebra 2
4-1 Matrices and Data
Does the dimension matter?
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Holt Algebra 2
4-1 Matrices and Data
matrix
dimensions
entry
address
scalar
Vocabulary
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Holt Algebra 2
4-1 Matrices and Data
matrix
dimensions
entry
address
scalar
Vocabulary
![Page 10: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/10.jpg)
Holt Algebra 2
4-1 Matrices and Data
Matrix A has two rows and three columns. A matrix with m rows and n columns has dimensions m n, read “m by n,” and is called an m n matrix. A has dimensions 2 3. Each value in a matrix is called an entry of the matrix.
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Holt Algebra 2
4-1 Matrices and Data
The address of an entry is its location in a matrix, expressed by using the lower case matrix letter with row and column number as subscripts. The score 16.206 is located in row 2 column 1, so a21 is 16.206.
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Holt Algebra 2
4-1 Matrices and Data
The prices for different sandwiches are presented at right.
Example 1: Displaying Data in Matrix Form
6 in 9 in
Roast beef $3.95 $5.95
Turkey $3.75 $5.60
Tuna $3.50 $5.25
A. Display the data in matrix form. P =
3.95 5.95
3.75 5.60
3.50 5.25
B. What are the dimensions of P?
P has three rows and two columns, so it is a 3 2
matrix.
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Holt Algebra 2
4-1 Matrices and Data
Example 1: Displaying Data in Matrix Form
C. What is entry P32? What does is represent?
D. What is the address of the entry 5.95?
The entry at P32, in row 3 column 2, is 5.25. It is the price of a 9 in. tuna sandwich.
The entry 5.95 is at P12.
The prices for different sandwiches are presented at right.
6 in 9 in
Roast beef $3.95 $5.95
Turkey $3.75 $5.60
Tuna $3.50 $5.25
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Holt Algebra 2
4-1 Matrices and Data
Use matrix M to answer the questions below.
Check It Out! Example 1
a. What are the dimensions of M? 3 4
11
m14 and m23
b. What is the entry at m32?
c. The entry 0 appears at what two addresses?
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Holt Algebra 2
4-1 Matrices and Data
You can add or subtract two matrices only if they have the same dimensions.
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Holt Algebra 2
4-1 Matrices and Data
Add or subtract, if possible.
Example 2A: Finding Matrix Sums and Differences
W + Y
W = , 3 –2
1 0 Y = ,
1 4
–2 3 X = ,
4 7 2
5 1 –1 Z =
2 –2 3
1 0 4
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Holt Algebra 2
4-1 Matrices and Data
X – Z
Example 2B: Finding Matrix Sums and Differences
W = , 3 –2
1 0 Y = ,
1 4
–2 3 X = ,
4 7 2
5 1 –1 Z =
2 –2 3
1 0 4
Add or subtract, if possible.
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Holt Algebra 2
4-1 Matrices and Data
X + Y
Example 2C: Finding Matrix Sums and Differences
W = , 3 –2
1 0 Y = ,
1 4
–2 3 X = ,
4 7 2
5 1 –1 Z =
2 –2 3
1 0 4
Add or subtract, if possible.
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Holt Algebra 2
4-1 Matrices and Data
Add or subtract if possible.
Check It Out! Example 2A
B + D
A = ,
4 –2
–3 10
2 6
B = , 4 –1 –5
3 2 8 C = ,
3 2
0 –9
–5 14
D = 0 1 –3
3 0 10
4 + 0 –1 + 1 –5 + (–3)
3 + 3 2 + 0 8 + 10
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Holt Algebra 2
4-1 Matrices and Data
B – A
Add or subtract if possible.
Check It Out! Example 2B
A = ,
4 –2
–3 10
2 6
B = , 4 –1 –5
3 2 8 C = ,
3 2
0 –9
–5 14
D = 0 1 –3
3 0 10
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Holt Algebra 2
4-1 Matrices and Data
D – B
Add or subtract if possible.
Check It Out! Example 2C
A = ,
4 –2
–3 10
2 6
B = , 4 –1 –5
3 2 8 C = ,
3 2
0 –9
–5 14
D = 0 1 –3
3 0 10
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Holt Algebra 2
4-1 Matrices and Data
Write a short summary to explain how to add matrices.
![Page 23: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/23.jpg)
Holt Algebra 2
4-1 Matrices and Data
Write a short summary to explain how to subtract matrices.
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Holt Algebra 2
4-1 Matrices and Data
4 –2
–3 10
2 6
A =
Scalar Multiplication
3A = ½ A =
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Holt Algebra 2
4-1 Matrices and Data
Write a short summary to explain how to Multiply a matrix by a scalar (constant) value.
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Holt Algebra 2
4-1 Matrices and Data
2x-5 4 = 25 4
3 3y+12 3 y+18
Evaluating equal matrices
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Holt Algebra 2
4-1 Matrices and Data
x+8 -5 = 38 -5
3 -y 3 4y-10
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Holt Algebra 2
4-1 Matrices and Data
x- 1 4 + -4 7 = 5 -1
-3 9 10 -2 6 3
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Holt Algebra 2
4-1 Matrices and Data
Example 4B: Simplifying Matrix Expressions
Evaluate 3R — P, if possible.
P =
3 –2
1 0
2 –1
Q= 4 7 2
5 1 –1 R =
1 4
–2 3
0 4
3(1) 3(4)
3(–2) 3(3)
3(0) 3(4)
–
3 –2
1 0
2 –1
3 12
–6 9
0 12
–
3 –2
1 0
2 –1
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Holt Algebra 2
4-1 Matrices and Data
Example 4A: Simplifying Matrix Expressions
Evaluate 3P — Q, if possible.
P =
3 –2
1 0
2 –1
Q= 4 7 2
5 1 –1 R =
1 4
–2 3
0 4
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Holt Algebra 2
4-1 Matrices and Data
Write a short summary to explain how to solve matrix equations.
![Page 32: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/32.jpg)
Holt Algebra 2
4-1 Matrices and Data
• The product of an m n and an n p matrix is an m p matrix.
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Holt Algebra 2
4-1 Matrices and Data
Let’s work through this together …
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Holt Algebra 2
4-1 Matrices and Data
Let’s work through this together …
![Page 35: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/35.jpg)
Holt Algebra 2
4-1 Matrices and Data
How do you finish the multiplication?
![Page 36: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/36.jpg)
Holt Algebra 2
4-1 Matrices and Data
How do you finish the multiplication?
![Page 37: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/37.jpg)
Holt Algebra 2
4-1 Matrices and Data
Find the products:
1. A2
![Page 38: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/38.jpg)
Holt Algebra 2
4-1 Matrices and Data
Find the products:
1. CB
![Page 39: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/39.jpg)
Holt Algebra 2
4-1 Matrices and Data
Find the products:
1. BA
![Page 40: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/40.jpg)
Holt Algebra 2
4-1 Matrices and Data
Find the products:
1. CD
![Page 41: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/41.jpg)
Holt Algebra 2
4-1 Matrices and Data
Find the products:
1. BA
![Page 42: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/42.jpg)
Holt Algebra 2
4-1 Matrices and Data
The product of two or more matrices is the matrix product. The following rules apply when multiplying matrices.
• The product of an m n and an n p matrix is an m p matrix.
![Page 43: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/43.jpg)
Holt Algebra 2
4-1 Matrices and Data
• The product of an m n and an n p matrix is an m p matrix.
![Page 44: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/44.jpg)
Holt Algebra 2
4-1 Matrices and Data
Tell whether the product is defined. If so, give its dimensions.
Example 1A: Identifying Matrix Products
A3 4 and B4 2; AB
A B AB
3 4 4 2 = 3 2 matrix
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Holt Algebra 2
4-1 Matrices and Data
Tell whether the product is defined. If so, give its dimensions.
Example 1B: Identifying Matrix Products
C1 4 and D3 4; CD
C D
1 4 3 4
The inner dimensions are not equal (4 ≠ 3), so the matrix product is not defined.
![Page 46: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/46.jpg)
Holt Algebra 2
4-1 Matrices and Data
Tell whether the product is defined. If so, give its dimensions.
P2 5 Q5 3 R4 3 S4 5
Check It Out! Example 1a
QP
![Page 47: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/47.jpg)
Holt Algebra 2
4-1 Matrices and Data
Tell whether the product is defined. If so, give its dimensions.
P2 5 Q5 3 R4 3 S4 5
Check It Out! Example 1b
SR
![Page 48: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/48.jpg)
Holt Algebra 2
4-1 Matrices and Data
Tell whether the product is defined. If so, give its dimensions.
P2 5 Q5 3 R4 3 S4 5
Check It Out! Example 1c
SQ
![Page 49: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/49.jpg)
Holt Algebra 2
4-1 Matrices and Data
Write a short summary to explain how to find matrix products.
![Page 50: Matrices and DataMatrices and Data - SharpSchool · Holt Algebra 2 44-1-1 Matrices and DataMatrices and Data ... spreadsheet as rows and columns of numbers. ... and real-world data](https://reader035.vdocuments.us/reader035/viewer/2022062908/5acecef97f8b9a4e7a8bd70d/html5/thumbnails/50.jpg)
Holt Algebra 2
4-1 Matrices and Data
1. What are the dimensions of A?
2. What is entry D12?
Evaluate if possible.
3. A — C 4. C + D 5. B + D