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x + y + 2z = 0 size 12{x+y+2z=0} {}
x + 2y + z = 0 size 12{x+2y+z=0} {}
2x + 3y + 3z = 0 size 12{2x+3y+3z=0} {}
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Find three solutions to the following system of equations.

x + 2y + z = 12 size 12{x+2y+z="12"} {}
y = 3 size 12{y=3} {}

(5, 3, 1), (4, 3, 2) (3, 3, 3)

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For what values of k size 12{k} {} the following system of equations have a). No solution? b). Infinitely many solutions?

x + 2y = 5 size 12{x+2y=5} {}
2x + 4y = k size 12{2x+4y=k} {}
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x + 3y z = 5 size 12{x+3y - z=5} {}

( 5 3s + t size 12{5 - 3s+t} {} , s size 12{s} {} , t size 12{t} {} )

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Why is it not possible for a linear system to have exactly two solutions? Explain geometrically.

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Inverse matrices

In the next two problems, verify that the given matrices are inverses of each other.

7 3 2 1 1 3 2 7 size 12{ left [ matrix { 7 {} # 3 {} ##2 {} # 1{} } right ]left [ matrix { 1 {} # - 3 {} ##- 2 {} # 7{} } right ]} {}
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1 1 0 1 0 1 2 3 4 3 4 1 2 4 1 3 5 1 size 12{ left [ matrix { 1 {} # - 1 {} # 0 {} ##1 {} # 0 {} # - 1 {} ## 2 {} # 3 {} # - 4{}} right ] left [ matrix {3 {} # - 4 {} # 1 {} ## 2 {} # - 4 {} # 1 {} ##3 {} # - 5 {} # 1{} } right ]} {}
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In the following problems, find the inverse of each matrix by the row-reduction method.

3 5 1 2 size 12{ left [ matrix { 3 {} # - 5 {} ##- 1 {} # 2{} } right ]} {}
2 5 1 3 size 12{ left [ matrix { 2 {} # 5 {} ##1 {} # 3{} } right ]} {}
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1 0 2 0 1 4 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} # 2 {} ##0 {} # 1 {} # 4 {} ## 0 {} # 0 {} # 1{}} right ]} {}
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1 1 1 1 0 1 2 1 1 size 12{ left [ matrix { 1 {} # 1 {} # - 1 {} ##1 {} # 0 {} # 1 {} ## 2 {} # 1 {} # 1{}} right ]} {}
1 2 –1 –1 –3 2 –1 –1 1 size 12{ left [ matrix { 1 {} # 2 {} # –1 {} ##–1 {} # –3 {} # 2 {} ## –1 {} # –1 {} # 1{}} right ]} {}
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1 1 1 3 1 0 1 1 2 size 12{ left [ matrix { 1 {} # 1 {} # 1 {} ##3 {} # 1 {} # 0 {} ## 1 {} # 1 {} # 2{}} right ]} {}
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In the following problems, first express the system as AX = B, and then solve using matrix inverses found in the preceding four problems.

3x 5y = 2 size 12{3x - 5y=2} {}
x + 2y = 0 size 12{ - x+2y=0} {}

(4, 2)

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x + 2z = 8 y + 4z = 8 z = 3 size 12{ matrix { x {} # +{} {} # {} # {} # 2z {} # ={} {} # 8 {} ##{} # {} # y {} # +{} {} # 4z {} # ={} {} # 8 {} ## {} # {} # {} # {} # z {} # ={} {} # 3{}} } {}
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x + y z = 2 x + z = 7 2x + y + z = 13 size 12{ matrix { x {} # +{} {} # y {} # - {} {} # z {} # ={} {} # 2 {} ##x {} # +{} {} # {} # {} # z {} # ={} {} # 7 {} ## 2x {} # +{} {} # y {} # +{} {} # z {} # ={} {} # "13"{}} } {}

(3, 3, 4)

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x + y + z = 2 3x + y = 7 x + y + 2z = 3 size 12{ matrix { x {} # +{} {} # y {} # +{} {} # z {} # ={} {} # 2 {} ##3x {} # +{} {} # y {} # {} # {} # ={} {} # 7 {} ## x {} # +{} {} # y {} # +{} {} # 2z {} # ={} {} # 3{}} } {}
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Why is it necessary that a matrix be a square matrix for its inverse to exist? Explain by relating the matrix to a system of equations.

If a matrix M size 12{M} {} has an inverse, then the system of linear equations that has M size 12{M} {} as its coefficient matrix has a unique solution. If a system of linear equations has a unique solution, then the number of equations must be the same as the number of variables. Therefore, the matrix that represents its coefficient matrix must be a square matrix.

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Suppose we are solving a system AX = B size 12{ ital "AX"=B} {} by the matrix inverse method, but discover A size 12{A} {} has no inverse. How else can we solve this system? What can be said about the solutions of this system?

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Application of matrices in cryptography

In the following problems, the letters A to Z correspond to the numbers 1 to 26, as shown below, and a space is represented by the number 27.

A B C D E F G H I J K L M
1 2 3 4 5 6 7 8 9 10 11 12 13
N O P Q R S T U V W X Y Z
14 15 16 17 18 19 20 21 22 23 24 25 26

In the next two problems, use the matrix A, given below, to encode the given messages.

A = 3 2 1 1 size 12{A= left [ matrix { 3 {} # 2 {} ##1 {} # 1{} } right ]} {}

In the two problems following, decode the messages that were encoded using matrix A.

Make sure to consider the spaces between words, but ignore all punctuation. Add a final space if necessary.

Encode the message: WATCH OUT!

71 24 66 23 78 35 87 36 114 47 size 12{ left [ matrix { "71" {} ##"24" } right ]left [ matrix { "66" {} ##"23" } right ]left [ matrix { "78" {} ##"35" } right ]left [ matrix { "87" {} ##"36" } right ]left [ matrix { "114" {} ##"47" } right ]} {}
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Encode the message: HELP IS ON THE WAY.

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Decode the following message:

64 23 102 41 82 32 97 35 71 28 69 32

RETURN HOME

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Decode the following message:

105 40 117 48 39 19 69 32 72 27 37 15 114 47

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In the next two problems, use the matrix B, given below, to encode the given messages.

B = 1 0 0 2 1 2 1 0 1 size 12{B= left [ matrix { 1 {} # 0 {} # 0 {} ##2 {} # 1 {} # 2 {} ## 1 {} # 0 {} # - 1{}} right ]} {}

In the two problems following, decode the messages that were encoded using matrix B size 12{B} {} .

Make sure to consider the spaces between words, but ignore all punctuation. Add a final space(s) if necessary.

Encode the message using matrix B size 12{B} {} :

LUCK IS ON YOUR SIDE.

12 51 9 11 67 2 19 95 14 14 105 11 15 87 3 27 91 18 4 67 23 size 12{ left [ matrix { "12" {} ##"51" {} ## 9} right ] left [ matrix {"11" {} ## "67" {} ##2 } right ]left [ matrix { "19" {} ##"95" {} ## "14"} right ] left [ matrix {"14" {} ## "105" {} ##- "11" } right ]left [ matrix { "15" {} ##"87" {} ## - 3} right ] left [ matrix {"27" {} ## "91" {} ##"18" } right ]left [ matrix { 4 {} ##"67" {} ## - "23"} right ]} {}
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Encode the message using matrix B size 12{B} {} :

MAY THE FORCE BE WITH YOU.

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Decode the following message that was encoded using matrix B:

8 23 7 4 47 –2 15 102 –12 20 58 15 27 80 18 12 74 –7

HEAD FOR THE HILLS

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Decode the following message that was encoded using matrix B:

12 69 –3 11 53 9 5 46 –10 18 95 –9 25 107 4 27 76 22 1 72 –26

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Questions & Answers

how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
William Reply
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
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At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?
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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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