MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr...
Transcript of MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr...
![Page 1: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/1.jpg)
MAS151: Civil EngineeringMathematics
Dr James [email protected]
Friday 25th October 2019, 12noonHicks Building LT1
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Course matters
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Your comments
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We are interested to know your opinions about this course:
usethe discussion board and the end of semester questionnaire.
If you have comments about your class tutor, please mentionthem by name.
![Page 5: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/5.jpg)
We are interested to know your opinions about this course: usethe discussion board and the end of semester questionnaire.
If you have comments about your class tutor, please mentionthem by name.
![Page 6: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/6.jpg)
We are interested to know your opinions about this course: usethe discussion board and the end of semester questionnaire.
If you have comments about your class tutor, please mentionthem by name.
![Page 7: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/7.jpg)
Reading week
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Week 7 (November 11–15) is a reading week.
There will be no classes or new videos released in readingweek: the videos released late in week 6 will be due in early inweek 8.
You should use the time to revise or catch up with thematerial so far, e.g by working on exercises.
![Page 9: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/9.jpg)
Week 7 (November 11–15) is a reading week.
There will be no classes or new videos released in readingweek: the videos released late in week 6 will be due in early inweek 8.
You should use the time to revise or catch up with thematerial so far, e.g by working on exercises.
![Page 10: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/10.jpg)
Week 7 (November 11–15) is a reading week.
There will be no classes or new videos released in readingweek: the videos released late in week 6 will be due in early inweek 8.
You should use the time to revise or catch up with thematerial so far, e.g by working on exercises.
![Page 11: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/11.jpg)
Matrices
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Later in this course (Semester 2) we will spend a good amountof time studying matrices.
However, they are so fundamentalto engineering mathematics that they may have alreadyappeared elsewhere in your course or could come up before weget to them. To help you to get comfortable in their use, wewill cover some of the basics today.
![Page 13: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/13.jpg)
Later in this course (Semester 2) we will spend a good amountof time studying matrices. However, they are so fundamentalto engineering mathematics that they may have alreadyappeared elsewhere in your course or could come up before weget to them.
To help you to get comfortable in their use, wewill cover some of the basics today.
![Page 14: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/14.jpg)
Later in this course (Semester 2) we will spend a good amountof time studying matrices. However, they are so fundamentalto engineering mathematics that they may have alreadyappeared elsewhere in your course or could come up before weget to them. To help you to get comfortable in their use, wewill cover some of the basics today.
![Page 15: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/15.jpg)
Why matrices?
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Matrices to model systems
A certain city consists of an urban area and suburbs.
Eachyear 5% of those living in the urban area move to the suburbsand 3% of those living in the suburbs move to the urban area.If there are initially 600, 000 people in the urban area and400, 000 in the suburbs, how many are in each 25 years later?
![Page 17: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/17.jpg)
Matrices to model systems
A certain city consists of an urban area and suburbs. Eachyear 5% of those living in the urban area move to the suburbs
and 3% of those living in the suburbs move to the urban area.If there are initially 600, 000 people in the urban area and400, 000 in the suburbs, how many are in each 25 years later?
![Page 18: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/18.jpg)
Matrices to model systems
A certain city consists of an urban area and suburbs. Eachyear 5% of those living in the urban area move to the suburbsand 3% of those living in the suburbs move to the urban area.
If there are initially 600, 000 people in the urban area and400, 000 in the suburbs, how many are in each 25 years later?
![Page 19: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/19.jpg)
Matrices to model systems
A certain city consists of an urban area and suburbs. Eachyear 5% of those living in the urban area move to the suburbsand 3% of those living in the suburbs move to the urban area.If there are initially 600, 000 people in the urban area and400, 000 in the suburbs, how many are in each 25 years later?
![Page 20: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/20.jpg)
Definitions
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Let m and n be positive integers.
Then an m× n matrix A isan array of real numbers, with m rows and n columns; that is
A =
a11 a12 . . . a1na21 a22 . . . a2n
......
. . ....
am1 am2 . . . amn
We sometimes write A = (aij) for the above matrix.
![Page 22: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/22.jpg)
Let m and n be positive integers. Then an m× n matrix A isan array of real numbers, with m rows and n columns;
that is
A =
a11 a12 . . . a1na21 a22 . . . a2n
......
. . ....
am1 am2 . . . amn
We sometimes write A = (aij) for the above matrix.
![Page 23: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/23.jpg)
Let m and n be positive integers. Then an m× n matrix A isan array of real numbers, with m rows and n columns; that is
A =
a11 a12 . . . a1na21 a22 . . . a2n
......
. . ....
am1 am2 . . . amn
We sometimes write A = (aij) for the above matrix.
![Page 24: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/24.jpg)
Let m and n be positive integers. Then an m× n matrix A isan array of real numbers, with m rows and n columns; that is
A =
a11 a12 . . . a1na21 a22 . . . a2n
......
. . ....
am1 am2 . . . amn
We sometimes write A = (aij) for the above matrix.
![Page 25: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/25.jpg)
Let m and n be positive integers. Then an m× n matrix A isan array of real numbers, with m rows and n columns; that is
A =
a11 a12 . . . a1na21 a22 . . . a2n
......
. . ....
am1 am2 . . . amn
We sometimes write A = (aij) for the above matrix.
![Page 26: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/26.jpg)
For example,
(3 1 20 2 1
)is a 2× 3 matrix, and
2√2 7
0 1
3 4−√2
7 7
is a 4× 2 matrix.
![Page 27: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/27.jpg)
For example,(3 1 20 2 1
)
is a 2× 3 matrix, and
2√2 7
0 1
3 4−√2
7 7
is a 4× 2 matrix.
![Page 28: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/28.jpg)
For example,(3 1 20 2 1
)is a 2× 3 matrix,
and
2√2 7
0 1
3 4−√2
7 7
is a 4× 2 matrix.
![Page 29: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/29.jpg)
For example,(3 1 20 2 1
)is a 2× 3 matrix, and
2√2 7
0 1
3 4−√2
7 7
is a 4× 2 matrix.
![Page 30: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/30.jpg)
For example,(3 1 20 2 1
)is a 2× 3 matrix, and
2√2 7
0 1
3 4−√2
7 7
is a 4× 2 matrix.
![Page 31: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/31.jpg)
For example,(3 1 20 2 1
)is a 2× 3 matrix, and
2√2 7
0 1
3 4−√2
7 7
is a 4× 2 matrix.
![Page 32: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/32.jpg)
Identity matrices
Let n be a positive integer.
Then the n× n matrix In given by
In =
1 0 . . . 00 1 . . . 0...
.... . .
...0 0 . . . 1
is called the identity matrix of size n.
![Page 33: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/33.jpg)
Identity matrices
Let n be a positive integer. Then the n× n matrix In given by
In =
1 0 . . . 00 1 . . . 0...
.... . .
...0 0 . . . 1
is called the identity matrix of size n.
![Page 34: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/34.jpg)
Identity matrices
Let n be a positive integer. Then the n× n matrix In given by
In =
1 0 . . . 00 1 . . . 0...
.... . .
...0 0 . . . 1
is called the identity matrix of size n.
![Page 35: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/35.jpg)
Identity matrices
Let n be a positive integer. Then the n× n matrix In given by
In =
1 0 . . . 00 1 . . . 0...
.... . .
...0 0 . . . 1
is called the identity matrix of size n.
![Page 36: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/36.jpg)
For example,
I2 =
(1 00 1
)and I3 =
1 0 00 1 00 0 1
.
The identity matrix In is always square. That is, it has thesame number of rows and columns.
![Page 37: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/37.jpg)
For example,
I2 =
(1 00 1
)
and I3 =
1 0 00 1 00 0 1
.
The identity matrix In is always square. That is, it has thesame number of rows and columns.
![Page 38: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/38.jpg)
For example,
I2 =
(1 00 1
)and I3 =
1 0 00 1 00 0 1
.
The identity matrix In is always square. That is, it has thesame number of rows and columns.
![Page 39: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/39.jpg)
For example,
I2 =
(1 00 1
)and I3 =
1 0 00 1 00 0 1
.
The identity matrix In is always square.
That is, it has thesame number of rows and columns.
![Page 40: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/40.jpg)
For example,
I2 =
(1 00 1
)and I3 =
1 0 00 1 00 0 1
.
The identity matrix In is always square. That is, it has thesame number of rows and columns.
![Page 41: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/41.jpg)
Matrix operations
![Page 42: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/42.jpg)
To use matrices we have to define some basic mathematicaloperations.
Two basic operations are matrix addition andmatrix multiplication.
![Page 43: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/43.jpg)
To use matrices we have to define some basic mathematicaloperations. Two basic operations are matrix addition
andmatrix multiplication.
![Page 44: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/44.jpg)
To use matrices we have to define some basic mathematicaloperations. Two basic operations are matrix addition andmatrix multiplication.
![Page 45: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/45.jpg)
Matrix addition
Let A = (aij) and B = (bij) both be m× n matrices.
Thenwe define the sum of A and B by
A+B =
a11 + b11 a12 + b12 . . . a1n + b1na21 + b21 a22 + b22 . . . a2n + b2n
......
. . ....
am1 + bm1 am2 + bm2 . . . amn + bmn
.
In other words, to add two matrices of the same dimensionssimply add their entries componentwise.
![Page 46: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/46.jpg)
Matrix addition
Let A = (aij) and B = (bij) both be m× n matrices. Thenwe define the sum of A and B by
A+B =
a11 + b11 a12 + b12 . . . a1n + b1na21 + b21 a22 + b22 . . . a2n + b2n
......
. . ....
am1 + bm1 am2 + bm2 . . . amn + bmn
.
In other words, to add two matrices of the same dimensionssimply add their entries componentwise.
![Page 47: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/47.jpg)
Matrix addition
Let A = (aij) and B = (bij) both be m× n matrices. Thenwe define the sum of A and B by
A+B =
a11 + b11 a12 + b12 . . . a1n + b1na21 + b21 a22 + b22 . . . a2n + b2n
......
. . ....
am1 + bm1 am2 + bm2 . . . amn + bmn
.
In other words, to add two matrices of the same dimensionssimply add their entries componentwise.
![Page 48: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/48.jpg)
Matrix addition
Let A = (aij) and B = (bij) both be m× n matrices. Thenwe define the sum of A and B by
A+B =
a11 + b11 a12 + b12 . . . a1n + b1na21 + b21 a22 + b22 . . . a2n + b2n
......
. . ....
am1 + bm1 am2 + bm2 . . . amn + bmn
.
In other words, to add two matrices of the same dimensions
simply add their entries componentwise.
![Page 49: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/49.jpg)
Matrix addition
Let A = (aij) and B = (bij) both be m× n matrices. Thenwe define the sum of A and B by
A+B =
a11 + b11 a12 + b12 . . . a1n + b1na21 + b21 a22 + b22 . . . a2n + b2n
......
. . ....
am1 + bm1 am2 + bm2 . . . amn + bmn
.
In other words, to add two matrices of the same dimensionssimply add their entries componentwise.
![Page 50: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/50.jpg)
For example,(1 0 00 1 0
)+
(2 0 34 2 0
)=
(3 0 34 3 0
).
![Page 51: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/51.jpg)
For example,(1 0 00 1 0
)+
(2 0 34 2 0
)=
(3 0 34 3 0
).
![Page 52: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/52.jpg)
Warning!
It is not possible to add two matrices if their dimensions aredifferent, so take care!
![Page 53: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/53.jpg)
Matrix multiplication
![Page 54: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/54.jpg)
A big reason why matrices are so useful comes down to therule for how they multiply.
Matrices can only be multiplied when the dimensions matchup in the right way.
The thing to remember is that thenumber of columns of the first matrix must be the same as thenumber of rows of the second one.
That is, if A is p× q and B is q × r, then we can find theirproduct. The result, AB, is a p× r matrix.
![Page 55: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/55.jpg)
A big reason why matrices are so useful comes down to therule for how they multiply.
Matrices can only be multiplied when the dimensions matchup in the right way. The thing to remember is that thenumber of columns of the first matrix must be the same as thenumber of rows of the second one.
That is, if A is p× q and B is q × r, then we can find theirproduct. The result, AB, is a p× r matrix.
![Page 56: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/56.jpg)
A big reason why matrices are so useful comes down to therule for how they multiply.
Matrices can only be multiplied when the dimensions matchup in the right way. The thing to remember is that thenumber of columns of the first matrix must be the same as thenumber of rows of the second one.
That is, if A is p× q and B is q × r,
then we can find theirproduct. The result, AB, is a p× r matrix.
![Page 57: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/57.jpg)
A big reason why matrices are so useful comes down to therule for how they multiply.
Matrices can only be multiplied when the dimensions matchup in the right way. The thing to remember is that thenumber of columns of the first matrix must be the same as thenumber of rows of the second one.
That is, if A is p× q and B is q × r, then we can find theirproduct.
The result, AB, is a p× r matrix.
![Page 58: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/58.jpg)
A big reason why matrices are so useful comes down to therule for how they multiply.
Matrices can only be multiplied when the dimensions matchup in the right way. The thing to remember is that thenumber of columns of the first matrix must be the same as thenumber of rows of the second one.
That is, if A is p× q and B is q × r, then we can find theirproduct. The result, AB, is a p× r matrix.
![Page 59: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/59.jpg)
We illustrate the procedure with an example.
Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 60: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/60.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 61: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/61.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A
(starting from the top)and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 62: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/62.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)
and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 63: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/63.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B
(starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 64: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/64.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B (starting from theleft)
in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 65: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/65.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 66: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/66.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 67: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/67.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2
1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 68: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/68.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.0
0.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 69: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/69.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2
0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 70: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/70.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 71: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/71.jpg)
We illustrate the procedure with an example. Let
A =
(1 2 30 1 1
)and B =
2 03 42 0
To find AB, we take each row from A (starting from the top)and ‘multiply it’ by each column from B (starting from theleft) in the following way:
AB =
(1.2 + 2.3 + 3.2 1.0 + 2.4 + 3.00.2 + 1.3 + 1.2 0.0 + 1.4 + 1.0
)=
(14 85 4
).
![Page 72: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/72.jpg)
In the previous example, A is 2× 3 and B is 3× 2
and theresult is 2× 2. Of course, BA will not be the same matrix, asthe result will be 3× 3.
![Page 73: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/73.jpg)
In the previous example, A is 2× 3 and B is 3× 2 and theresult is 2× 2.
Of course, BA will not be the same matrix, asthe result will be 3× 3.
![Page 74: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/74.jpg)
In the previous example, A is 2× 3 and B is 3× 2 and theresult is 2× 2. Of course, BA will not be the same matrix,
asthe result will be 3× 3.
![Page 75: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/75.jpg)
In the previous example, A is 2× 3 and B is 3× 2 and theresult is 2× 2. Of course, BA will not be the same matrix, asthe result will be 3× 3.
![Page 76: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/76.jpg)
Column vectors
One case that occurs frequently is when the second matrix is acolumn vector (i.e. an n× 1 matrix) of a suitable length.
Forexample,
(1 3 12 0 −1
) 322
=
(1.3 + 3.2 + 1.2
2.3 + 0.2 + (−1).2
)=
(114
).
The result will always be a column vector, although not ingeneral with the same length.
![Page 77: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/77.jpg)
Column vectors
One case that occurs frequently is when the second matrix is acolumn vector (i.e. an n× 1 matrix) of a suitable length. Forexample,
(1 3 12 0 −1
) 322
=
(1.3 + 3.2 + 1.2
2.3 + 0.2 + (−1).2
)=
(114
).
The result will always be a column vector, although not ingeneral with the same length.
![Page 78: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/78.jpg)
Column vectors
One case that occurs frequently is when the second matrix is acolumn vector (i.e. an n× 1 matrix) of a suitable length. Forexample,
(1 3 12 0 −1
) 322
=
(1.3 + 3.2 + 1.2
2.3 + 0.2 + (−1).2
)=
(114
).
The result will always be a column vector, although not ingeneral with the same length.
![Page 79: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/79.jpg)
Column vectors
One case that occurs frequently is when the second matrix is acolumn vector (i.e. an n× 1 matrix) of a suitable length. Forexample,
(1 3 12 0 −1
) 322
=
(1.3 + 3.2 + 1.2
2.3 + 0.2 + (−1).2
)=
(114
).
The result will always be a column vector, although not ingeneral with the same length.
![Page 80: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/80.jpg)
Column vectors
One case that occurs frequently is when the second matrix is acolumn vector (i.e. an n× 1 matrix) of a suitable length. Forexample,
(1 3 12 0 −1
) 322
=
(1.3 + 3.2 + 1.2
2.3 + 0.2 + (−1).2
)=
(114
).
The result will always be a column vector, although not ingeneral with the same length.
![Page 81: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/81.jpg)
Column vectors
One case that occurs frequently is when the second matrix is acolumn vector (i.e. an n× 1 matrix) of a suitable length. Forexample,
(1 3 12 0 −1
) 322
=
(1.3 + 3.2 + 1.2
2.3 + 0.2 + (−1).2
)=
(114
).
The result will always be a column vector,
although not ingeneral with the same length.
![Page 82: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/82.jpg)
Column vectors
One case that occurs frequently is when the second matrix is acolumn vector (i.e. an n× 1 matrix) of a suitable length. Forexample,
(1 3 12 0 −1
) 322
=
(1.3 + 3.2 + 1.2
2.3 + 0.2 + (−1).2
)=
(114
).
The result will always be a column vector, although not ingeneral with the same length.
![Page 83: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/83.jpg)
Another thing to notice is that multiplication by the identitymatrix
(of the correct size) will leave the other matrixunchanged. For example,(
1 00 1
)(1 3 12 0 −1
)=
(1.1 + 0.2 1.3 + 0.0 1.1 + 0.(−1)0.1 + 1.2 0.3 + 1.0 0.1 + 1.(−1)
)=
(1 3 12 0 −1
).
![Page 84: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/84.jpg)
Another thing to notice is that multiplication by the identitymatrix (of the correct size)
will leave the other matrixunchanged. For example,(
1 00 1
)(1 3 12 0 −1
)=
(1.1 + 0.2 1.3 + 0.0 1.1 + 0.(−1)0.1 + 1.2 0.3 + 1.0 0.1 + 1.(−1)
)=
(1 3 12 0 −1
).
![Page 85: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/85.jpg)
Another thing to notice is that multiplication by the identitymatrix (of the correct size) will leave the other matrixunchanged.
For example,(1 00 1
)(1 3 12 0 −1
)=
(1.1 + 0.2 1.3 + 0.0 1.1 + 0.(−1)0.1 + 1.2 0.3 + 1.0 0.1 + 1.(−1)
)=
(1 3 12 0 −1
).
![Page 86: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/86.jpg)
Another thing to notice is that multiplication by the identitymatrix (of the correct size) will leave the other matrixunchanged. For example,
(1 00 1
)(1 3 12 0 −1
)=
(1.1 + 0.2 1.3 + 0.0 1.1 + 0.(−1)0.1 + 1.2 0.3 + 1.0 0.1 + 1.(−1)
)=
(1 3 12 0 −1
).
![Page 87: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/87.jpg)
Another thing to notice is that multiplication by the identitymatrix (of the correct size) will leave the other matrixunchanged. For example,(
1 00 1
)(1 3 12 0 −1
)=
(1.1 + 0.2 1.3 + 0.0 1.1 + 0.(−1)0.1 + 1.2 0.3 + 1.0 0.1 + 1.(−1)
)
=
(1 3 12 0 −1
).
![Page 88: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/88.jpg)
Another thing to notice is that multiplication by the identitymatrix (of the correct size) will leave the other matrixunchanged. For example,(
1 00 1
)(1 3 12 0 −1
)=
(1.1 + 0.2 1.3 + 0.0 1.1 + 0.(−1)0.1 + 1.2 0.3 + 1.0 0.1 + 1.(−1)
)=
(1 3 12 0 −1
).
![Page 89: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/89.jpg)
Activity. Working in groups of two or three, find a matrix Asuch that
A
(xurban
xsuburban
)=
(0.95xurban + 0.03xsuburban
0.05xurban + 0.97xsuburban
).
![Page 90: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/90.jpg)
A =
(0.95 0.030.05 0.97
).
In the example at the beginning of the lecture,
A
(600, 000400, 000
)will give the number of people in the urban and suburbanareas after one year. Multiplying by A repeatedly means thepopulations after 25 years will be given by
A25
(600, 000400, 000
).
![Page 91: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/91.jpg)
A =
(0.95 0.030.05 0.97
).
In the example at the beginning of the lecture,
A
(600, 000400, 000
)will give the number of people in the urban and suburbanareas after one year.
Multiplying by A repeatedly means thepopulations after 25 years will be given by
A25
(600, 000400, 000
).
![Page 92: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/92.jpg)
A =
(0.95 0.030.05 0.97
).
In the example at the beginning of the lecture,
A
(600, 000400, 000
)will give the number of people in the urban and suburbanareas after one year. Multiplying by A repeatedly means thepopulations after 25 years will be given by
A25
(600, 000400, 000
).
![Page 93: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/93.jpg)
And finally. . .
![Page 94: MAS151: Civil Engineering Mathematics · 2020. 1. 30. · MAS151: Civil Engineering Mathematics Dr James Cranch mas-engineering@sheffield.ac.uk Friday 25th October 2019, 12noon Hicks](https://reader035.fdocuments.net/reader035/viewer/2022062509/60f1059de193565d4f1699c1/html5/thumbnails/94.jpg)
Reminders:
• email address [email protected]
• website http://engmaths.group.shef.ac.uk/mas151
(also accessible through MOLE).