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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

(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
Shahid Reply
bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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