<< Chapter < Page Chapter >> Page >

Given the matrices E size 12{E} {} , F size 12{F} {} , G size 12{G} {} and H size 12{H} {} , below

E = 1 2 4 2 3 1 size 12{E= left [ matrix { 1 {} # 2 {} ##4 {} # 2 {} ## 3 {} # 1{}} right ]} {} F = 2 1 3 2 size 12{F= left [ matrix { 2 {} # - 1 {} ##3 {} # 2{} } right ]} {} G = 4 1 size 12{G= left [ matrix { 4 {} # 1{}} right ]} {} H = 3 1 size 12{H= left [ matrix { - 3 {} ##- 1 } right ]} {}

Find, if possible.

  1. EF size 12{ ital "EF"} {}
  2. FE size 12{ ital "FE"} {}
  3. FH size 12{ ital "FH"} {}
  4. GH size 12{ ital "GH"} {}
  1. To find EF size 12{ ital "EF"} {} , we multiply the first row 1 2 size 12{ left [ matrix { 1 {} # 2{}} right ]} {} of E size 12{E} {} with the columns 2 3 size 12{ left [ matrix { 2 {} ##3 } right ]} {} and 1 2 size 12{ left [ matrix { - 1 {} ##2 } right ]} {} of the matrix F size 12{F} {} , and then repeat the process by multiplying the other two rows of E size 12{E} {} with these columns of F size 12{F} {} . The result is as follows:

    EF = 1 2 4 2 3 1 2 1 3 2 = 1 2 + 2 3 1 1 + 2 2 4 2 + 2 3 4 1 + 2 2 3 2 + 1 3 3 1 + 1 2 = 8 3 14 0 9 1 size 12{ matrix { ital "EF" {} # ={} {} # left [ matrix {1 {} # 2 {} ## 4 {} # 2 {} ##3 {} # 1{} } right ]left [ matrix { 2 {} # - 1 {} ##3 {} # 2{} } right ]{} ## {} # ={} {} # left [ matrix {1 cdot 2+2 cdot 3 {} # 1 cdot - 1+2 cdot 2 {} ## 4 cdot 2+2 cdot 3 {} # 4 cdot - 1+2 cdot 2 {} ##3 cdot 2+1 cdot 3 {} # 3 cdot - 1+1 cdot 2{} } right ]= left [ matrix { 8 {} # 3 {} ##"14" {} # 0 {} ## 9 {} # - 1{}} right ]{} } } {}
  2. The product FE size 12{ ital "FE"} {} is not possible because the matrix F size 12{F} {} has two entries in each row, while the matrix E size 12{E} {} has three entries in each column. In other words, the matrix F size 12{F} {} has two columns, while the matrix E size 12{E} {} has three rows.

  3. FH = 2 1 3 2 3 1 = 2 3 + 1 1 3 3 + 2 1 = 5 11 size 12{ ital "FH"= left [ matrix { 2 {} # - 1 {} ##3 {} # 2{} } right ]left [ matrix { - 3 {} ##- 1 } right ]= left [ matrix { 2 cdot - 3+ - 1 cdot - 1 {} ##3 cdot - 3+2 cdot - 1 } right ]= left [ matrix { - 5 {} ##- "11" } right ]} {}

  4. GH = 4 1 3 1 = 4 3 + 1 1 = 13 size 12{ ital "GH"= left [ matrix { 4 {} # 1{}} right ] left [ matrix {- 3 {} ## - 1} right ]= left [4 cdot - 3+1 cdot - 1 right ]= left [ - "13" right ]} {}

Got questions? Get instant answers now!
Got questions? Get instant answers now!

We summarize matrix multiplication as follows:

In order for product AB size 12{ ital "AB"} {} to exist, the number of columns of A size 12{A} {} , must equal the number of rows of B size 12{B} {} . If matrix A size 12{A} {} is of dimension m × n size 12{m times n} {} and B size 12{B} {} of dimension n × p size 12{n times p} {} , the product will have the dimension m × p size 12{m times p} {} . Furthermore, matrix multiplication is not commutative.

Given the matrices R size 12{R} {} , S size 12{S} {} , and T size 12{T} {} below.

R = 1 0 2 2 1 5 2 3 1 size 12{R= left [ matrix { 1 {} # 0 {} # 2 {} ##2 {} # 1 {} # 5 {} ## 2 {} # 3 {} # 1{}} right ]} {} S = 0 1 2 3 1 0 4 2 1 size 12{S= left [ matrix { 0 {} # - 1 {} # 2 {} ##3 {} # 1 {} # 0 {} ## 4 {} # 2 {} # 1{}} right ]} {} T = 2 3 0 3 2 2 1 1 0 size 12{T= left [ matrix { - 2 {} # 3 {} # 0 {} ##- 3 {} # 2 {} # 2 {} ## - 1 {} # 1 {} # 0{}} right ]} {}

Find 2 RS 3 ST size 12{2 ital "RS" - 3 ital "ST"} {} .

We multiply the matrices R size 12{R} {} and S size 12{S} {} .

RS = 8 3 4 23 9 9 13 3 5 size 12{ ital "RS"= left [ matrix { 8 {} # 3 {} # 4 {} ##"23" {} # 9 {} # 9 {} ## "13" {} # 3 {} # 5{}} right ]} {}
2 RS = 2 8 3 4 23 9 9 13 3 5 = 16 6 8 46 18 18 26 6 10 size 12{2 ital "RS"=2 left [ matrix { 8 {} # 3 {} # 4 {} ##"23" {} # 9 {} # 9 {} ## "13" {} # 3 {} # 5{}} right ]= left [ matrix {"16" {} # 6 {} # 8 {} ## "46" {} # "18" {} # "18" {} ##"26" {} # 6 {} # "10"{} } right ]} {}
ST = 1 0 2 9 11 2 15 17 4 size 12{ ital "ST"= left [ matrix { 1 {} # 0 {} # - 2 {} ##- 9 {} # "11" {} # 2 {} ## - "15" {} # "17" {} # 4{}} right ]} {}
3 ST = 3 1 0 2 9 11 2 15 17 4 = 3 0 6 27 33 6 45 51 12 size 12{3 ital "ST"=3 left [ matrix { 1 {} # 0 {} # - 2 {} ##- 9 {} # "11" {} # 2 {} ## - "15" {} # "17" {} # 4{}} right ]= left [ matrix {3 {} # 0 {} # - 6 {} ## - "27" {} # "33" {} # 6 {} ##- "45" {} # "51" {} # "12"{} } right ]} {}
2 RS 3 ST = 16 6 8 46 18 18 26 6 10 3 0 6 27 33 6 45 51 12 = 13 6 14 73 15 12 71 45 2 size 12{2 ital "RS" - 3 ital "ST"= left [ matrix { "16" {} # 6 {} # 8 {} ##"46" {} # "18" {} # "18" {} ## "26" {} # 6 {} # "10"{}} right ] - left [ matrix {3 {} # 0 {} # - 6 {} ## - "27" {} # "33" {} # 6 {} ##- "45" {} # "51" {} # "12"{} } right ]= left [ matrix { "13" {} # 6 {} # "14" {} ##"73" {} # - "15" {} # "12" {} ## "71" {} # - "45" {} # - 2{}} right ]} {}
Got questions? Get instant answers now!
Got questions? Get instant answers now!

In this chapter, we will be using matrices to solve linear systems. In [link] , we will be asked to express linear systems as the matrix equation AX = B size 12{ ital "AX"=B} {} , where A size 12{A} {} , X size 12{X} {} , and B size 12{B} {} are matrices. The matrix A size 12{A} {} is called the coefficient matrix .

Verify that the system of two linear equations with two unknowns:

ax + by = h size 12{ ital "ax"+ ital "by"=h} {}
cx + dy = k size 12{ ital "cx"+ ital "dy"=k} {}

can be written as AX = B size 12{ ital "AX"=B} {} , where

A = a b c d size 12{A= left [ matrix { a {} # b {} ##c {} # d{} } right ]} {} X = x y size 12{X= left [ matrix { x {} ##y } right ]} {} and   B = h k size 12{B= left [ matrix { h {} ##k } right ]} {}

If we multiply the matrices A size 12{A} {} and X size 12{X} {} , we get

AX = a b c d x y = ax + by cx + dy size 12{ ital "AX"= left [ matrix { a {} # b {} ##c {} # d{} } right ]left [ matrix { x {} ##y } right ]= left [ matrix { ital "ax"+ ital "by" {} ##ital "cx"+ ital "dy" } right ]} {}

If AX = B size 12{ ital "AX"=B} {} then

ax + by cx + dy = h k size 12{ left [ matrix { ital "ax"+ ital "by" {} ##ital "cx"+ ital "dy" } right ]= left [ matrix { h {} ##k } right ]} {}

If two matrices are equal, then their corresponding entries are equal. Therefore, it follows that

ax + by = h size 12{ ital "ax"+ ital "by"=h} {}
cx + dy = k size 12{ ital "cx"+ ital "dy"=k} {}
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Express the following system as AX = B size 12{ ital "AX"=B} {} .

2x + 3y 4z = 5 size 12{2x+"3y"–4z=5} {}
3x + 4y 5z = 6 size 12{3x+4y - 5z=6} {}
5x 6z = 7 size 12{5x - 6z=7} {}

The above system of equations can be expressed in the form AX = B size 12{ ital "AX"=B} {} as shown below.

2 3 4 3 4 5 5 0 6 x y z = 5 6 7 size 12{ left [ matrix { 2 {} # 3 {} # - 4 {} ##3 {} # 4 {} # - 5 {} ## 5 {} # 0 {} # - 6{}} right ] left [ matrix {x {} ## y {} ##z } right ]= left [ matrix { 5 {} ##6 {} ## 7} right ]} {}
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Systems of linear equations; gauss-jordan method

In this section, we learn to solve systems of linear equations using a process called the Gauss-Jordan method. The process begins by first expressing the system as a matrix, and then reducing it to an equivalent system by simple row operations. The process is continued until the solution is obvious from the matrix. The matrix that represents the system is called the augmented matrix , and the arithmetic manipulation that is used to move from a system to a reduced equivalent system is called a row operation .

Write the following system as an augmented matrix.

2x + 3y 4z = 5 size 12{2x+3y - 4z=5} {}
3x + 4y 5z = 6 size 12{3x+4y - 5z= - 6} {}
4x + 5y 6z = 7 size 12{4x+5y - 6z=7} {}

We express the above information in matrix form. Since a system is entirely determined by its coefficient matrix and by its matrix of constant terms, the augmented matrix will include only the coefficient matrix and the constant matrix. So the augmented matrix we get is as follows:

2 3 4 5 3 4 5 6 4 5 6 7 size 12{ left [ matrix { 2 {} # 3 {} # - 4 {} # \lline {} # 5 {} ##3 {} # 4 {} # - 5 {} # \lline {} # - 6 {} ## 4 {} # 5 {} # - 6 {} # \lline {} # 7{}} right ]} {}
Got questions? Get instant answers now!
Got questions? Get instant answers now!

In the [link] , we expressed the system of equations as AX = B size 12{ ital "AX"=B} {} , where A size 12{A} {} represented the coefficient matrix, and B size 12{B} {} the matrix of constant terms. As an augmented matrix, we write the matrix as A B size 12{ left [A \lline B right ]} {} . It is clear that all of the information is maintained in this matrix form, and only the letters x size 12{x} {} , y size 12{y} {} and z size 12{z} {} are missing. A student may choose to write x size 12{x} {} , y size 12{y} {} and z size 12{z} {} on top of the first three columns to help ease the transition.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold
Logan Reply
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?
Chine Reply
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?
Chalton Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Applied finite mathematics' conversation and receive update notifications?

Ask