review solve the system of equations. 1 2, -1, 1
TRANSCRIPT
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Review
Solve the system of equations.
1
2
22
23
zyx
zyx
zyx
2, -1, 1
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Matrices
2
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Definition
• Some Words: One: Matrix
More than one: Matrices • Definition: In Mathematics, matrices are used
to store information.• This information is written in a rectangular
arrangement of rows and columns.• Each entry, or element, of a matrix has a
precise position and meaning.3
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Example
• Food shopping online: people go online to shop for items and have them delivered to their homes.
• Cartons of eggs, bread, packets of vegetables, bags of rice, packets of fish were ordered online and the people left their address for delivery.
• A selection of orders may look like this:
4
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ExampleOrder
Address
Carton of eggs
bread vegetables rice fish
10 Kros
Road
0 2 2 2 1
15 Usmar St
0 2 1 1 3
17 High St 1 2 1 0 0
22 Ofar Rd.
4 0 0 1 35
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Example
• The dispatch people will be interested in the numbers:
This is a 4 by 5 matrix
0 2 2 2 1
0 2 1 1 3
1 2 1 0 0
4 0 0 1 3
4 rows
5 columns 6
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Definition
A matrix is defined by its order which is always number of rows by number of columns
7
R X C
2 rows
3 columns
2 X 3 matrix
2 5 8
1 6 1
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Exercise• Consider the network below showing the
roads connecting four towns and the distances, in km, along each road.
8
A
14
C
D
B5
10
8
12
16
(i) Write down the information in matrix form. (ii) What is the order of the matrix?
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Solution(i) This information could be put into a table:
9
km
A B C D
A 0 5 14 12
B 5 0 10 16
C 14 10 0 8
D 12 16 8 0
to
from
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Solutionand then into a matrix:
10
0 5 14 12
5 0 10 16
14 10 0 8
12 16 8 0
(ii) order: R X C = 4 X 4 matrix.This is called a square matrix.
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Definition
A square matrix has the same number of rows as columns. Its order is of the form M x M.
Examples:
11
1 0
0 1
2 X 2 square matrix
2 0 6
3 5 18
7 8 3
3 X 3 square matrix
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DefinitionThe transpose of a matrix M, called MT, is found by interchanging the rows and columns.
Example: M =
12
2 3
7 9
2
3
7
9
rowrow
column
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Definition
Equal Matrices: Two matrices are equal if theircorresponding entries (elements) are equal.
Example: If
13
a b
c d
10 2
4 8
a = 10
c = 4 d = 8
b = -2
=
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DefinitionEntries, or elements, of a matrix are named according to their position in the matrix. The row is named first and the column second.
Example: entry a23 is the element on row 2, column 3.
Example: here are the entries for a 2 x 2 matrix.
14
11 12
21 22
a a
a a
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ExampleIn the following matrix, name the position of the colored entry.(i)
15
1-752
Remember: row firsta2
Column second
row 2
column 1The entry is a21
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Example
In the following matrix, name the position of the colored entry.
(ii)
16
c d e f
o p q r
row 1, column 3
The entry is a13
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Example• In the following matrices, identify the value of
the entry for the given position.
17
7 8
2 1
3 5
7 5 3 0
10 9 0 2
1 0 5 11
a32
a24
row 3, column 2
= 5
row 2, column 4= 2
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College Kaitlin wants to attend one of three Iowauniversities next year. She has gathered informationabout tuition (T), room and board (R/B), andenrollment (E) for the universities. Use a matrix toorganize the information. Which university’s total cost is lowest?
Iowa State University:T - $3132 R/B - $4432 E - 26,845
University of Iowa:T - $3204 R/B - $4597 E - 28,311
University of Northern Iowa:T - $3130 R/B - $4149 E - 14,106
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Organize the data into labeled columns and rows.
ISU
UI
UNI
T R/B E
Answer: The University of Northern Iowa has the lowest total cost.
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Dining Out Justin is going out for lunch. The information he has gathered from the two fast-food restaurants is listed below. Use a matrix to organize the information. When is each restaurant’s total cost less expensive?
Burger Complex Lunch Express
Hamburger Meal $3.39 Hamburger Meal $3.49
Cheeseburger Meal $3.59 Cheeseburger Meal $3.79
Chicken Sandwich Meal $4.99 Chicken Sandwich Meal $4.89
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The Burger Complex has the best price for hamburgers and cheeseburgers. Lunch Express has the best price for chicken sandwiches.
Answer:
Burger Complex
Lunch Express
HamburgerMeal
Cheese-burgerMeal
ChickenSandwich
Meal
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State the dimensions of matrix G if
4 columns
2 rows
Answer: Since matrix G has 2 rows and 4 columns, the dimensions of
matrix G are 2 4.
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State the dimensions of matrix G if
Answer: 3 2
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Solve for x and y.
Since the matrices are equal, the corresponding elements are equal. When you write the sentences to solve this equation, two linear equations are formed.
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Second equation
This system can be solved using substitution.
Substitute 3x – 2 for y.
Distributive Property
Add 4 to each side.
Divide each side by 7.
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To find the value for y, substitute 1 for x in either equation.
Substitute 1 for x.
First equation
Simplify.
Answer: The solution is (1, 1).
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Solve for x and y.
Answer: (2, 5)
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Review
State the dimensions of each matrix. Then identify the position of the circled element in each matrix.
28
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Review
Solve the equation.
29
935332 yzx
12
15
5
4
y
xx
35
5
4
512 x
yyx
yx