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Algebraic

In the following exercises, show that matrix A is the inverse of matrix B .

A = [ 1 0 −1 1 ] , B = [ 1 0 1 1 ]

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A = [ 1 2 3 4 ] , B = [ −2 1 3 2 1 2 ]

A B = B A = [ 1 0 0 1 ] = I

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A = [ 4 5 7 0 ] , B = [ 0 1 7 1 5 4 35 ]

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A = [ −2 1 2 3 −1 ] , B = [ −2 −1 −6 −4 ]

A B = B A = [ 1 0 0 1 ] = I

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A = [ 1 0 1 0 1 −1 0 1 1 ] , B = 1 2 [ 2 1 −1 0 1 1 0 −1 1 ]

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A = [ 1 2 3 4 0 2 1 6 9 ] , B = 1 4 [ 6 0 −2 17 −3 −5 −12 2 4 ]

A B = B A = [ 1 0 0 0 1 0 0 0 1 ] = I

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A = [ 3 8 2 1 1 1 5 6 12 ] , B = 1 36 [ −6 84 −6 7 −26 1 −1 −22 5 ]

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For the following exercises, find the multiplicative inverse of each matrix, if it exists.

[ 3 −2 1 9 ]

1 29 [ 9 2 −1 3 ]

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[ −3 7 9 2 ]

1 69 [ −2 7 9 3 ]

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[ 1 1 2 2 ]

There is no inverse

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[ 0.5 1.5 1 −0.5 ]

4 7 [ 0.5 1.5 1 −0.5 ]

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[ 1 0 6 −2 1 7 3 0 2 ]

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[ 0 1 −3 4 1 0 1 0 5 ]

1 17 [ −5 5 −3 20 −3 12 1 −1 4 ]

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[ 1 2 −1 −3 4 1 −2 −4 −5 ]

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[ 1 9 −3 2 5 6 4 −2 7 ]

1 209 [ 47 −57 69 10 19 −12 −24 38 −13 ]

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[ 1 −2 3 −4 8 −12 1 4 2 ]

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[ 1 2 1 2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 ]

[ 18 60 −168 −56 −140 448 40 80 −280 ]

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For the following exercises, solve the system using the inverse of a 2 × 2 matrix.

   5 x 6 y = 61 4 x + 3 y = 2

( −5 , 6 )

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8 x + 4 y = −100 3 x −4 y = 1

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3 x −2 y = 6 x + 5 y = −2

( 2 , 0 )

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5 x −4 y = −5 4 x + y = 2.3

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−3 x −4 y = 9 12 x + 4 y = −6

( 1 3 , 5 2 )

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−2 x + 3 y = 3 10 x + 5 y = 1 2

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8 5 x 4 5 y = 2 5 8 5 x + 1 5 y = 7 10

( 2 3 , 11 6 )

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1 2 x + 1 5 y = 1 4 1 2 x 3 5 y = 9 4

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For the following exercises, solve a system using the inverse of a 3 × 3 matrix.

3 x −2 y + 5 z = 21 5 x + 4 y = 37 x −2 y −5 z = 5

( 7 , 1 2 , 1 5 )

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   4 x + 4 y + 4 z = 40    2 x 3 y + 4 z = −12   x + 3 y + 4 z = 9

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    6 x 5 y z = 31   x + 2 y + z = −6   3 x + 3 y + 2 z = 13

( 5 , 0 , −1 )

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6 x −5 y + 2 z = −4 2 x + 5 y z = 12 2 x + 5 y + z = 12

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4 x −2 y + 3 z = −12 2 x + 2 y −9 z = 33 6 y −4 z = 1

1 34 ( −35 , −97 , −154 )

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1 10 x 1 5 y + 4 z = −41 2 1 5 x −20 y + 2 5 z = −101 3 10 x + 4 y 3 10 z = 23

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1 2 x 1 5 y + 1 5 z = 31 100 3 4 x 1 4 y + 1 2 z = 7 40 4 5 x 1 2 y + 3 2 z = 1 4

1 690 ( 65 , −1136 , −229 )

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0.1 x + 0.2 y + 0.3 z = −1.4 0.1 x −0.2 y + 0.3 z = 0.6 0.4 y + 0.9 z = −2

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Technology

For the following exercises, use a calculator to solve the system of equations with matrix inverses.

2 x y = −3 x + 2 y = 2.3

( 37 30 , 8 15 )

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1 2 x 3 2 y = 43 20 5 2 x + 11 5 y = 31 4

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12.3 x −2 y −2.5 z = 2 36.9 x + 7 y −7.5 z = −7 8 y −5 z = −10

( 10 123 , −1 , 2 5 )

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0.5 x −3 y + 6 z = −0.8 0.7 x −2 y = −0.06 0.5 x + 4 y + 5 z = 0

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Extensions

For the following exercises, find the inverse of the given matrix.

[ 1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 ]

1 2 [ 2 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 ]

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[ 1 0 2 5 0 0 0 2 0 2 1 0 1 3 0 1 ]

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[ 1 2 3 0 0 1 0 2 1 4 2 3 5 0 1 1 ]

1 39 [ 3 2 1 7 18 53 32 10 24 36 21 9 9 46 16 5 ]

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[ 1 2 0 2 3 0 2 1 0 0 0 0 3 0 1 0 2 0 0 1 0 0 1 2 0 ]

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[ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 ]

[ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 ]

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Real-world applications

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.

2,400 tickets were sold for a basketball game. If the prices for floor 1 and floor 2 were different, and the total amount of money brought in is $64,000, how much was the price of each ticket?

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In the previous exercise, if you were told there were 400 more tickets sold for floor 2 than floor 1, how much was the price of each ticket?

Infinite solutions.

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A food drive collected two different types of canned goods, green beans and kidney beans. The total number of collected cans was 350 and the total weight of all donated food was 348 lb, 12 oz. If the green bean cans weigh 2 oz less than the kidney bean cans, how many of each can was donated?

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Students were asked to bring their favorite fruit to class. 95% of the fruits consisted of banana, apple, and oranges. If oranges were twice as popular as bananas, and apples were 5% less popular than bananas, what are the percentages of each individual fruit?

50% oranges, 25% bananas, 20% apples

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A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $1 and the chocolate chip cookies at $0.75. They raised $700 and sold 850 items. How many brownies and how many cookies were sold?

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A clothing store needs to order new inventory. It has three different types of hats for sale: straw hats, beanies, and cowboy hats. The straw hat is priced at $13.99, the beanie at $7.99, and the cowboy hat at $14.49. If 100 hats were sold this past quarter, $1,119 was taken in by sales, and the amount of beanies sold was 10 more than cowboy hats, how many of each should the clothing store order to replace those already sold?

10 straw hats, 50 beanies, 40 cowboy hats

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Anna, Ashley, and Andrea weigh a combined 370 lb. If Andrea weighs 20 lb more than Ashley, and Anna weighs 1.5 times as much as Ashley, how much does each girl weigh?

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Three roommates shared a package of 12 ice cream bars, but no one remembers who ate how many. If Tom ate twice as many ice cream bars as Joe, and Albert ate three less than Tom, how many ice cream bars did each roommate eat?

Tom ate 6, Joe ate 3, and Albert ate 3.

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A farmer constructed a chicken coop out of chicken wire, wood, and plywood. The chicken wire cost $2 per square foot, the wood $10 per square foot, and the plywood $5 per square foot. The farmer spent a total of $51, and the total amount of materials used was 14  ft 2 . He used 3 ft 2 more chicken wire than plywood. How much of each material in did the farmer use?

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Jay has lemon, orange, and pomegranate trees in his backyard. An orange weighs 8 oz, a lemon 5 oz, and a pomegranate 11 oz. Jay picked 142 pieces of fruit weighing a total of 70 lb, 10 oz. He picked 15.5 times more oranges than pomegranates. How many of each fruit did Jay pick?

124 oranges, 10 lemons, 8 pomegranates

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Questions & Answers

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
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discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
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Yousuf Reply
the study of living organisms and their interactions with one another and their environments
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the study of living organisms and their interactions with one another and their environment.
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discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
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cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
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Diversity of living thing
ISCONT
what is cell division
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Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
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life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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