Matrices and its Applications: Albert Alipan Tayong
Albert Alipan Tayong, LMT Albert Alipan Tayong, LMT Page 1
Matrices and its Applications: Albert Alipan Tayong
Introduction:
This article about matrices is dedicated to my students and fellow teachers in Majan College (University College).
May the lessons you are about to learn arrive in wonderful and applicable ways.
The scenarios, applications, exercises, word problems, and examples given herein are perceived universally. However, the names and representations are focusing more on the Sultanate of Oman and its capital-Muscat as I made these discussions and/or lessons here in this beautiful and peaceful country.
Furthermore, kindly note that almost all of the problems, exercises, and examples are carefully structured and represented using my creative imaginations, background, experiences, and knowledge. Some are recollections of my thoughts when I was still studying and working in various places, some are practice exercises given in my classes, which are linked to the teaching outline, and most are relevant problems derived by myself as a result of my innovative research and explorations. This means that the general content specified herein are very useful and powerful since it has already been used in real-life situations. In short, the sources specified at the end of this section have been used only as a basis and a point of reference. I have devoted much of my time editing and modifying the context in order to see to it that the contents are beneficial to all college students in all aspects. Related practice exercises are given in order to help students master the concepts, allowing them to solve such exercises independently. Answers have been double checked to avoid confusions on the part of the readers.
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Matrices and its Applications: Albert Alipan Tayong
PREFACE
Matrices are very important in many aspects. In the field of Information Technology, it helps students solve various problems using excel functions. One of the most important usages of matrices in computer side applications is the encryption of message of codes. Matrices and their inverses are used by programmers for coding or encrypting a message. A message is made as a sequence of numbers in a binary format for communication and it follows code theory for solving. Hence, with the help of matrices, those equations are solved. With these encryptions only, internet functions are working and even banks could work with a transmission of sensitive and private data. Also, in computer based applications, matrices play a vital role in the projection of three dimensional image into a two dimensional screen, creating the realistic seeming motions. Stochastic matrices and Eigen vector solvers are used in the page rank algorithms which are used in the ranking of web pages in Google search. The matrix calculus is used in the generalization of analytical notions like exponentials and derivatives to their higher dimensions. In physics, matrices are applied in the study of electrical circuits, quantum mechanics and optics. In the calculation of battery power outputs, resistor conversion of electrical energy into another useful energy, these matrices play a major role in calculations. Especially in solving problems using Kirchoff’s laws of voltage and current, matrices are essential. In geology, matrices are used for taking seismic surveys. They are used for plotting graphs, statistics and also to do scientific studies in almost different fields. Matrices are used in representing the real world data like the traits of people, habits, etc. They are the best representation methods for plotting the common survey things. Matrices are used in calculating the gross domestic products in economics which eventually helps in calculating the goods production efficiently. Matrices are used in many organizations such as for scientist for recording the data for their experiments. In robotics and automation, matrices are the base elements for the robot movements. The movements of the robots are programmed with the calculation of matrices’ rows and columns. The inputs for controlling robots are given based on the calculations from matrices. Page 3
Matrices and its Applications: Albert Alipan Tayong
This article aims to:
enable students to define a matrix, write the position of matrix elements, identify basic types of matrices, and recognize dimension of matrices. develop a mathematical foundation on matrices that is vital to problem solving. perform operations on matrices using excel and manual computations. use basic properties related to the operations on matrices to solve exercises and problems easily and accurately. solve word problems involving matrices and apply such techniques in real world. utilize basic knowledge and problem solving techniques on matrices in their future studies in various fields as mentioned above.
Acknowledgements: I would like to express my warmest thanks to all the students and staffs of the college where I am currently working. Special thanks are given to my students in Mathematical and Statistical Skills module who appreciated and valued so much my time, knowledge, skills, teaching methodologies and strategies. The feedback I got from you has challenged my intelligence and inspired me from the bottom of my heart. Also, I am indebted to all my students in Business Mathematics & Statistics (BMAS) in Business Department and Basic Mathematics and Applied Mathematics in Foundation Programme.
Above all, I would like to thank our God, Almighty who let this article come into fruition.
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Matrices and its Applications: Albert Alipan Tayong
CONTENTS
1. Definition of a Matrix 2. Position of Elements of a Matrix 3. Order/Dimension/Degree of Matrices 4. Basic Types of Matrices 5. Addition and Subtraction of Matrices 6. How to use a Scientific Calculator to Combine Fractional Terms. 7. Exercise A 8. Problems Involving Addition and Subtraction of Matrices 9. Properties of Matrix Addition and Subtraction 10. Scalar Multiplication and Multiplication of Matrices 11. Transpose of a Matrix 12. Exercise B 13. Inverse of a Matrix 14. Operations on Matrices using excel functions 15. Word Problems Involving Multiplication of Matrices, Cramer’s Rule, and Inverse of a Matrix 16. Exercise C 17. Answers, Page 15, Exercise A 18. Answers, Page 24, Exercise B 19. Answers, Page 49, Exercise C 20. Bibliography Page 5
Matrices and its Applications: Albert Alipan Tayong
1. Definition of a Matrix A matrix is a rectangular array of real numbers or variables such that each number or variable has a definite position allotted to it. And, for positioning purposes, all matrices always contain a row and a column. These numbers or variables are called elements of the matrix.
For example, A=
3
5
is a matrix with elements
7
2
3, 5, 7, 2, 4, and 1
4
1
Where, Row 1: 3
5
Row 2: 7
2
Row 3: 4
1
Column 1: 3
7
4
Column 2: 5
2
1
We usually use capital or big letters to denote a matrix and its elements are always enclosed with braces or brackets. However, braces are commonly used and the same will be applied in this section. In our example above, we used a capital letter A and enclosed the elements with braces. 2. Position of Elements of a Matrix For example, B=
3
5
a11 = 3; a12 = 5
7
2
a21 = 7; a22 = 2
4
1
a31 = 4; a32 = 1
Note: The position always starts from a11 and uses a small letter (a). Add 1 whenever you move to the right and add 10 whenever you go down.
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Matrices and its Applications: Albert Alipan Tayong
Thought process:
R1
C1 C2
a11= 3 means row 1, column 1
3
a12= 5 means row 1, column 2
5
a21= 7 means row 2, column 1 R2
7
2
a22= 2 means row 2, column 2 a31= 4 means row 3, column 1
R3
4
1
a32= 1 means row 3, column 2
Example 1: Find the position of the elements of the following matrix. C= C=
4
9
3
2
Solution: a11 = 4; a12 = 9 a21 = 3; a22 = 2
Example 2: Using the matrix below, write the elements and their corresponding positions.
A= A=
5
-3
0
2
1
7
Solution: a11 = 5; a12 = -3 a13 = 0 a21 = 2; a22 = 1
a23 = 7
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Matrices and its Applications: Albert Alipan Tayong
3. Order/Dimension/Degree of Matrices The dimension of any matrix (also called order or degree) tells us how many rows and how many columns a matrix has. A row is an array of numbers or variables from left to right. A column is a list of numbers or variables from top to bottom. Dimension: R x C Where: R = number of rows; C = number of columns For example,
6
10
13
1
Its dimension is 3 x 4.
2
3
0
7
That means matrix A has
4
2
2
5
3 rows and 4 columns.
A=
Example 1: Find the order of the following matrices: 5
-3
0
2
1
7
F= A=
AD==
5
-3
0
2
1
7
3
2
9
Order: 2 x 3
Order: 3 x 3
65 A= 1
Order: 3 x 1
9
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Matrices and its Applications: Albert Alipan Tayong
4. Basic Types of Matrices (1) Row Matrix – is a matrix which has exactly one row. For example, (9
2
3 8) is a row matrix.
(2) Column Matrix – is a matrix which has exactly one column. 11 For example,
18
is a column matrix.
3 (3) Square Matrix – is a matrix in which the number of rows is equal to the number of columns. For this reason, the dimension of any square matrix is C x C or N x N where C or N is the number of columns.
For example,
6 11 10
3 0 9
2 1 7
is a 3 x 3 square matrix.
Note: Square matrices have these dimensions: 1 x 1, 2 x 2, 3 x 3, … Other examples of square matrices are given below. [7],
1x1
5
3
1
9
2x2
,
2
4
3
15
2
5
4
2
3
1
9
0
1
3
1
7
1
11
8
6
7
5
5
6
10
3x3
,
4x4
NoteNote that,that the list matrices goesgoes on up 5 xto5,5 6x x5,6,6 and on the of listsquare of square matrices ontoup x 6,so and and so so on forth. this book this focuses 2 x 2 and square andNevertheless, so forth. Nevertheless, bookon focuses on 2 3x x2 3and 3x matrices because of its wide applications in real life. But, 1 x 1, 4 x 4, 3 square matrices because of its wide applications in real life. But,and 1 5 x 5x 1, square matrices will also be tackled in order to have something 4 x 4, and 5 x 5 square matrices will also be tackled in order toto use have whenever such situations presentsuch themselves in present real life.themselves This will be something to use whenever situations discussed with be excel 2007-2013inapplications. in realin conjunction life. This will discussed conjunction with excel applications. Page 9
Matrices and its Applications: Albert Alipan Tayong
(4) Diagonal Matrix- is a square matrix whose every element above and below the main diagonal elements is zero. 6 0 0
For example,
0 11 0
0 0 4
is a diagonal matrix.
The diagonal elements are 6, 11, and 4. You might have noticed that each and every element above and below this diagonal line is 0. Kindly bear in mind that the diagonal elements in a diagonal matrix can also be zero. For example, 3 0 0
0 0 0
0 0 12
,
40 0
0 0
0 0 0
, and
0 0 0
0 0 0
are also diagonal matrices because as illustrated above, all the elements or entries above and below the diagonal line are 0’s. (5) Scalar Matrix- is a diagonal matrix whose diagonal entries are equal. Examples: 4 0 0
0 4 0
0 0 4
,
10
0
0
10
0 0 0
,
0 0 0
0 0 0
,
1
0
0
1
As you can see above, all the matrices are square, all entries above and below the diagonal lines are 0’s, and all diagonal elements are equal. This means that the given matrices are all scalar since all satisfy the definition of a scalar matrix. Please also bear in mind that all the diagonal lines above are drawn just to emphasize the diagonal elements and hence; in listing and/or writing matrices, these diagonal lines are not needed. (6) Identity/Unit matrix- is a diagonal are all 1. For example, 1 1 0 0 0 1 and 0 Page 10
matrix whose diagonal elements
0 1 0
0 0 1
are identity matrices.
Matrices and its Applications: Albert Alipan Tayong
(7) Triangular matrix – is either lower triangular or upper triangular. Lower Triangular matrix – is a square matrix whose elements above the main diagonal elements are all zero.
i)
6 3 2
For example,
0 11 1
0 0 4
is a lower triangular matrix.
As you can see above, all the elements above the diagonal line are all zero and all the elements except those above the diagonal line form a lower triangle. For this reason, the matrix is lower triangular.
ii)
Upper Triangular matrix – is a square matrix whose elements below the main diagonal elements are all zero.
0 0 0
For example,
3 15 0
4 7 4
is an upper triangular matrix.
As you can see above, all the elements below the diagonal line are all zero and all the elements except those below the diagonal line form an upper triangle. For this reason, the matrix is upper triangular. Other examples of triangular matrices are given below. B=
4
0
3
2
6
7
0
2
C=
C
B is a lower triangular matrix.
C is an upper triangular matrix.
(8) Zero or Null Matrix- is any matrix whose elements or entries are all 0. For example, 0 0
0 0
0 0
0 0
0 0
and
Page 11
0
0
0
0
0
0
are null matrices.
Matrices and its Applications: Albert Alipan Tayong
5. Addition and Subtraction of Matrices In order to perform addition and subtraction, matrices must have the same dimension or order. Otherwise, it is undefined. Example 1: Find the sum.
A=
3 5 1
4 2 8
6 0 5
B=
2 -3 -3
10 4 -4
31 7 8
Solution:
A+B=
3+2 5 + (-3) 1 + (-3)
4 + 10 2+4 8+ (-4)
6 + 31 0+7 5+8
=
5 14 37 2 6 7 -2 4 13
Example 2: Find C + B, A – C, and B + D.
A=
2 3 1 1 0 4 7 8 4
B=
1 9 -4 3 8 10
C+B=
1+1 8+9 15 + 12 1 + (-4) 2 + 3 11 + 6 4+8 19 + 10 1 + 11
A-C=
2-1 1-1 7-4
3-8 0-2 8 -19
1 - 15 4-11 4 -1
12 6 11
C=
1 1 4
8 15 2 11 19 1
=
2 17 -3 5 12 29
=
1 -5 -14 0 -2 -7 3 -11 3
7 D= 6
27 17 12
B + D is undefined because the dimensions of the matrices are not equal. B is a 3 x 3 matrix and D is a 2 x 1 matrix.
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Matrices and its Applications: Albert Alipan Tayong
Example 3: Find A + 0, A + D, B – A, and A + B.
1
A=
2
3
1
0
1 13
1
11
B=
3
-4
4
5
9
3
5
D=
1
1
8
1
2
Solutions:
1
A+0=A=
2
3
1
0
1 3 3
1
4
A + D is undefined because the dimensions of the matrices are not equal. A is 2 x 3 and D is 2 x 2.
B–A=
1-
1
11
2
5
-4 - 1
1 A+B=
2
+1
1 + (-4)
-3
9-1 1
3-0
3+
5
11 5
0+3
-
1 3 4
1 3 3
=
4
1 3 +
+9 1 5
=
1
−4
23
2
5
-5
3
3
26
31
2
5
-3
3
3 19
3 −11 20
20
Note: Addition and subtraction of matrices are only applicable if the matrices have the same orders. Adding and subtracting are done by pairing the elements (e.g. a11 + a11, a12 + a12 or a11 – a11, a12 – a12 and so on and so forth.) Adding and subtracting whole numbers and integers are easier compared to adding fractions and mixed numbers. Use a scientific calculator whenever the exercises involve fractions or mixed numbers. Kindly note that all throughout this booklet, answers on matrices involving fractions must be expressed in fractions and not in decimals in order to have exact answers, save space, and avoid confusion. Page 13
Matrices and its Applications: Albert Alipan Tayong
6. How to use a scientific calculator to combine fractional terms?
All fractions must be enclosed with parentheses. Otherwise, the calculator will display a wrong answer. Final answers involving fractions must be in its simplest forms. This can be done by pressing shift a b/c. Other calculators are automatically converting the answers into simplest fractions. If you have such calculators, there is no need to do this step.
For example,
1
-
5
3 4
In the calculator, press the open and close parentheses and then put the values. Kindly note that all fraction bars must be converted into (÷). 1 3 Hence, becomes (1÷ 5) - (3÷ 4) = 5 4 The calculator will then display the answer after pressing the equal sign (=). It displays the value, which is equal to -0.55. After which, press SHIFT a b/c. The answer will then be converted into a simplest fraction number. Hence, we have the following: Whole numbers must be separated without parentheses.
(1÷ 5) - (3÷ 4) = (1÷ 5) - (3÷ 4) = - 0.55 -0.55 SHIFT a b/c =
−11
Another example, 3
20 1 2
+
1
Calculator: 3 + (1 ÷ 2) + (1 ÷ 6) =
6
The calculator will then display this: 3.6666666666… 3.6666666666… SHIFT a b/c =
11
3 Remark: The answer is a non-terminating decimal. It never stops. For this reason, the best way to express this number is by changing it into its fractional form. In this manner we can save space and we will have exact answer as mentioned above. Page 14
Matrices and its Applications: Albert Alipan Tayong
7. Exercise A 1. Find the sum.
D=
4 4 8 5 3 -1 2 18 12
1 0 -1
F=
9 44 -2
3 11 17
2. Find M - Y, N + M, and B + N.
1 2 6 3 1 7 5 4 3
M=
2 5 7
N=
8 22 -3 10 7 12
6 B = -1 4
7 10 3 5 12 3
Y=
7 1 6
3
8
0
1
2
6
3. Find A + A, B + C, B – A, and A - B.
3 5 1
A=
2 7
7 3
8
B=
8
3
2
8
5
6
-1
7
11
3
4. Find the difference.
D=
4 4 8 5 3 -1 2 18 12
F=
1 0 -1
9 44 -2
3 11 17
5. Find F, when B + F = 0 or Null Matrix.
B=
4 5 2
4 8 3 -1 18 12
F=
Page 15
_ _ _
_ _ _
_ _ _
C=
Matrices and its Applications: Albert Alipan Tayong
8. Problems Involving Addition and Subtraction of Matrices Example 1: An IT company has two branches. In each branch, there are three Offices. In each office, there are 4 Accountants, 5 clerks, and 3 Managers. In one office of a branch, 9 salesmen are also working. In each office of the second branch, 2 HR Officers are also working. Using a matrix notation, find i)
the total number of posts of each kind in all offices taken together in each branch. the total number of posts of each kind in all the offices taken together from both branches. the remaining total number of staffs of each kind if the last office is closed and all the employees in this particular office are terminated.
ii) iii)
Solution: Let 1 and 2 = Branch 1 and Branch 2 A, B, C, D, E, and F = Offices A = Accountants C = Clerks M = Managers S = Salesmen H = HR Officers
1
A A = (4 B = (4 C = (4
C 5 5 5
2
D = (4 5 E = (4 5 F = (4 5
M 3 3 3
S 9 0 0
H 0) 0) 0)
3 0 2) 3 0 2) 3 0 2) A
i)
C M S H
Branch 1 = A + B + C = (12 15 9
9 0)
Branch 2 = D + E + F = (12 15
0 6)
9
ii)
All Offices = Branch 1 + Branch 2 = (24 30 18 9 6)
iii)
(24 30 18 9 6) – (4 5
3 0 2) = ( 20 25 15 9 4) Page 16
Matrices and its Applications: Albert Alipan Tayong
Example 2: Mr. Ahmed Al Shukairi’s IT Infrastructure has 3 accounts in Bank Muscat and 3 accounts in NBO namely: Dirhams, Euros, and Rials. The Bank statements are given below: Bank Muscat Records: In year 1, it has a balance of 5 million Dirhams, 1 million Euros, and 4 million Rials. In year 2, it has a balance of 1.2 million Dirhams, 0.4 million Euros, and 3 million Rials. NBO Records: In year 1, it has a balance of 0.8 million Dirhams, 0.1 million Euros, and 2 million Rials. In year 2, it has a balance of 0.5 million Dirhams, 0.3 million Euros, and 2 million Rials. i)
ii)
Given the data above, find the balance of each account taken together from both banks. Separate year 1 and year 2 and use matrix notations to solve the problem. Using the resulting matrix, find the total balance of each account combining year 1 and year 2.
Solution: Let D = Dirham Account E = Euro Account R = Rial Account
Bank Muscat: D
E
R
Year 1:
5
1
4
Year 2:
1.2
0.4
3
D
E
R
Year 1:
0.8
0.1
2
Year 2:
0.5
0.3
2
NBO:
D
E
R
5.8
1.1
6
1.7
0.7
5
i) Bank Muscat + NBO = ii) Total balance in year 1 and year 2 = (7.5 Page 17
1.8 11)
Matrices and its Applications: Albert Alipan Tayong
9. Properties of Matrix Addition and Subtraction i) Matrix addition is commutative. That is, A + B = B + A. For example,
A=
-4 5 7 4 2 -1 4 11 10
A+B=
-4 6 3
B=
13 9 42 9 9 26
0 2 -1
8 40 -2
B+A=
2 10 16 -4 6 3
13 9 42 9 9 26
As you can see above, A + B = B + A. Hence, we have proven that matrix addition is commutative. ii) Matrix subtraction is not commutative. That is, A – B ≠ B – A. For example,
A-B=
-4 -3 2 -38 5 13
5 -11 -6
B-A=
4 3 -5 -2 38 11 -5 -13 6
As you can see above, the elements of A – B are not equal with the elements of B – A. Therefore, A – B ≠ B – A. Hence, we have proven that matrix subtraction is not commutative. iii) Matrix addition is associative. That is, A + (B+ C) = (A + B) + C For example, A = (3 45 1), B = (2 9 11), C = (5 3 7) (A + B) + C = (5 54 12) + (5 3 7) = (10 57 19) A + (B + C) = (3 45 1) + (7 12 18) = (10 57 19) As you can see above, A + (B+ C) = (A + B) + C. Hence, we have proven that matrix addition is associative.
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Matrices and its Applications: Albert Alipan Tayong
10. Scalar Multiplication & Multiplication of Matrices Multiplying matrices can easily be done using the dimensions of matrices being multiplied. Left – to – right order must be followed because matrix multiplication in general is not commutative. That is, AB ≠ BA. Example 1: Find 2A * B.
AA==
5
-33.5
0
2
1
7
3
2
9
2A is an example of a scalar multiplication. The scalar quantity is 2 and it is multiplied to each of the elements.
65 B= 1 9 Solution:
2A A ==
2A * B =
2(5)
2(-33.5) 2(0)
2(2)
2(1)
2(7)
2(3)
2(2)
2(9)
10 4 6
-67 2 4 (3 x 3)
=
0 14 18
x
x
10
-67
0
4
2
14
6
4
18
65 1 9 (3 x 1)
Equal Dimension of the answer Page 19
Matrices and its Applications: Albert Alipan Tayong
2A * B is then a 3 x 1 matrix. That means that the matrix we need to find contains 3 rows and 1 column.
2A * B =
-
=
a11 a21 a31
Thought process: Make 3 blank rows, 1 blank column. As you can see above, the blank lines indicate that it is a 3 x 1 matrix. It is blank because in the beginning, we do not know the elements. We need to find them. For that we need to allot their positions. Always start with a11. Add 1 to the right if there is any, and add 10 whenever you move down. As you can see, it has only 1 column. So, we only have a11, plus 10 it becomes a21 down and so on. In multiplying matrices, we always follow this format below: a11 (means row 1 of the 1st matrix and column 1 of the second matrix) a21 (means row 2 of the 1st matrix and column 1 of the second matrix) a31 (means row 3 of the 1st matrix and column 1 of the second matrix) 1st 2nd
2A * B =
10 4 6
-67 2 4
0 14 18
x
65 1 9
a11 = 10(65) + (-67)(1) + 0(9) = 583 a21= 4(65) + 2(1) + 14(9) = 388 a31= 6(65) + 4(1) + 18(9) = 556 Hence,
2A * B =
583 388 556
Note: The middle values of the dimensions of the matrices must be equal. Otherwise, there is no answer. The arrows above allow you to visualize how the pairing is done. Always add the products of the values. Page 20
Matrices and its Applications: Albert Alipan Tayong
Example 2: Find CF and FC. 3 7
C=
CF =
3 7
1 4
1 4
2 8
F=
2 8
X
(2 X 3)
9 5
6 3
9 5
6 3
(2 X 2)
Not equal Therefore, CF is undefined. There is no answer because the middle values are not equal. This means that not all elements have partners. FC =
9 5
6 3
3 7
X
(2 X 2)
1 4
2 8
(2 X 3)
Equal Dimension of the answer
FC2x3 =
-
-
a11 = 9(3) + 6(7) = 69 a12 = 9(1) + 6(4) = 33 a13 = 9(2) + 6(8) = 66 a21 = 5(3) + 3(7) = 36 a22 = 5(1) + 3(4) = 17 a23 = 5(2) + 3(8) = 34
-
=
a11 a21
a12 a22
FC =
69 36
Page 21
a13 a23
33 17
66 34
Matrices and its Applications: Albert Alipan Tayong
As you can see above, CF is undefined and FC is a 2 x 3 matrix. Hence, CF ≠FC. Therefore, we have proven that matrix multiplication is not commutative. đ?&#x;? đ??‚ Example 3: Find B, 3A + B, and . đ?&#x;‘ đ?&#x;‘ 2
6
5
A=
B= 3
4
3
7
3(2) 3(3)
3A =
3A + B =
1/3(8)
3(6) 3(4)
6 9
=
18 12
6 9
2.2
,
2
½
4
5/3
1/3
7/3
8/3
18 12
5 7
+
1 1
1
3 4
3
( )
1
= 3C= 3
1 3
3
5
= 1/3(7)
C
8
1/3(1)
B=
C
Âź C=
,
1/3(5) 1
1
(2)
1 8
=
1 (2.2) 3
(5)
1 1
1
3 2
3
( )
1
5
11
12
3
15
2
1
4
3
6
3
=
Page 22
(4)
11 16
19 20
Matrices and its Applications: Albert Alipan Tayong
11. Transpose of a Matrix The transpose of matrix A is a matrix whose column elements are the row elements of matrix A. Note: To get the transpose, change the rows to columns. For example,
A=
3
2
6
4
7
9
2
5
8
Transpose of A = AT =
3
4
2
2
7
5
6
9
8
Another example, T T
Find 2B T and (C ) . B = (3
T
B =
T T
5
3 5 2
2),
,
(C ) = C =
3 2 6
C=
1 3 7
6 10 4
T
2B =
3
1
2 6
3 7
Kindly note that, (AT )T or (A′ )′ = A. You can use AT or A′ as a symbol for A Transpose.
Page 23
Matrices and its Applications: Albert Alipan Tayong
12. Exercise B 1. Mr. Smith has a company of 3 offices. Each Office is assigned 1 Manager and 2 Accountants. In addition, the first office has 1 HR Officer, 2 Architects, 3 Engineers, 2 Admin Officers, 4 Designers, 2 Secretaries, and 2 Cleaners. The second and the third Office have 1 HR Officer, 3 Architect, 4 Engineers, 4 Marketing Officers, 6 Designers, 3 Secretaries, 5 Salesmen, and 3 Cleaners. Using a matrix notation, find the following: a) the total number of posts of each kind in all offices taken together. b) the total number of employees.
2. Find AB, BA, K T , and ¾ C.
A= A=
2 0 9 6
K= K=
8 2/5 6 12
5 1 8 7
3 -1 10 4
2 3 2
B= B=
1/3 1/7 8 15
C=
-1/2 4 7
4 3 9
11 2 6
5 3 -4
-1/2 40 62
3. Given the matrices below, prove that A + B = B + A and C - D ≠ D - C. 2
6
5
A=
B= 3
D=
1
4
¼
5
2.2
2
½
4
C=
,
7
5
10
8
2
¾
14
8
Page 24
,
Matrices and its Applications: Albert Alipan Tayong
13. Inverse of a Matrix Only square matrices have inverses. For this reason, all other matrices which are not square have no inverses. Square matrices are 1 x 1, 2 x 2, 3 x 3, etc. For example,
A=
C=
3
2
6
4
7
9
2
5
3 7
1 4
8
2 8
is 3 x 3 square matrix. Therefore, it has an inverse
is a 2 x 3 matrix. It is not a square matrix, so it has no inverse.
Example 1: Find the inverse of A = (3) Solution: Thought process: A is a 1 x 1 matrix. It is square, so it has an inverse. This particular matrix is “trivialâ€? because it has a single element. Kindly note that to find the inverse of a matrix, all we need at the outset is its determinant. The determinant of any 1 x 1 matrix is equal to its element. Det A = | A | = 3 Since it is 1 x 1 and it contains only one entry, its inverse is its reciprocal. Hence, đ??´âˆ’1 =
1 3
Remark: The symbol of matrix inverse is A−1 , B −1 , etc, depending on the letter used.
Page 25
Matrices and its Applications: Albert Alipan Tayong
Example 2: Find the inverse of 4
1
3
7
A=
Solution: The above is a 2 x 2 matrix. The inverse of any 2 x 2 matrix can be found using the below formula. Kindly take note that this formula is designed by the author himself for easy solving. There are other formulas published on the internet and books. Those are also useful but in this section, we will be using the following:
A−1 =
a22
1 Det
∗
-a21
-a12 a11
Kindly note that a21 and a12 are always negative. Determinant = Product of the main diagonal elements – Product of the right diagonal elements. Step 1: Write the elements with their corresponding positions: a11 = 4 a21 = 3
a12 = 1 a22 = 7
Step 2: Find the determinant. Det A = 4(7) – 1(3) = 25 Step 3: Use the formula:
A−1 =
A−1 =
a22
1 Det
1 25
∗
-a21 7
-a12 a11 -1
∗
= -3
4
7
−1
25
25
−3
4
25
25
Page 26
Matrices and its Applications: Albert Alipan Tayong
Example 3: Find the inverse of -3
-2
7
5
F=
Solution: a11 = -3 a21 = 7
a12 = -2 a22 = 5
Det F = -3(5) – (-2)(7) = -1
F −1 =
F −1 =
1 Det
1 −1
F −1 = −1
F
−1
a22
∗
-a12
-a21
∗
∗
-5
-2
7
3
a11
5
2
-7
-3
5
2
-7
-3
=
Remark: We use F −1 because we are solving for the inverse of matrix F. The negatives in the formula change the signs of the elements. Above, you can see that -2 becomes 2 and 7 becomes -7. Please bear this in mind and never forget it. Once, you make a mistake here, your answer will be definitely wrong. Always remember that in Mathematics, negatives matter. And, we need to be vigilant with negatives all the time. Silly mistakes with negatives lead to disaster in your solutions. Always review your work and learn from your mistakes. Page 27
Matrices and its Applications: Albert Alipan Tayong
Example 4: Find the inverse of
M=
3
2
6
4
7
9
2
5
8
Solution: M is a 3 x 3 matrix. Step 1: Find the determinant. Use only column 1. Follow the signs below.
3 4 2
+3
7 5
For a 3 x 3 matrix, these signs must be followed at all times.
+ +
9 8
3 4 2
2 7 5
Encircle the 1st number in the 1st column and draw lines from left to right and top to bottom. Then use the remaining elements, which form a 2 x 2 matrix.
6 9 8
3[7(8) – 5(9)] = 3[56-45] = 3[11] = 33
-4
2 5
6 8
3 4 2
2 7 5
6 9 8
-4[2(8) – 6(5)] = -4[16-30] = -4[-14] = 56
+2
2 7
6 9
3 4 2 2
2 7 5
6 9 8
Get the determinant of the 2 x 2 matrix. The operation in the middle whenever we get the determinant is (-)
2[2(9) – 6(7)] = 2[18-42] = 2[-24] = -48 Add all the answers to get Det M.
Det M = 33 + 56 + (-48) = 41 Page 28
Matrices and its Applications: Albert Alipan Tayong
Step 2: Find the co-factors.
Co-factors =
+
-
+
-
+
-
+
-
+
a11 =
a21 a31
a12
a13
a22
a23
a32
a33
a11: +
7 5
9 8
3 4 2
2 7 5
6 9 8
2 7 5
6 9 8
a11 = [7(8) – 9(5)] = 11
a12: -
4 2
9 8
3 4 2
a12 = - [4(8) – 2(9)] = - [32-18) = -14
a13: +
4 2
7 5
3 4 2
2 7 5
6 9 8
a13 = [4(5) – 7(2)] = [20-14] = 6 Remark: Above, you might have noticed that we only used signs and there are no numbers beside them. This is always the case whenever we get the cofactors. Students sometimes forget this and as a result, all the answers are wrong. Kindly note that the lines drawn from left to right and top to bottom are optional. If you can mentally identify the elements of the 2 x 2 matrix, you may do it. After all, we are only concerned about each determinant, using the four remaining elements every time we delete the numbers on the line. Page 29
Matrices and its Applications: Albert Alipan Tayong
a21: -
2 5
6 8
3 4 2
2 7 5
6 9 8
a21 = - [2(8) – 5(6)] = -[16-30] = 14
a22: +
3 2
6 8
3 4 2
2 7 5
6 9 8
a22 = [3(8) – 2(6)] = [24-12] = 12
a23: -
3 2
2 5
3 4 2
2 7 5
6 9 8
a23 = - [3(5) – 2(2)] = - [15 – 4] = -11
a31: +
2 7
6 9
3 4 2
2 7 5
6 9 8
a31 = [2(9) – 6(7)] = [18 - 42) = -24
a32: -
3 4
6 9
3 4 2
2 7 5
6 9 8
a32 = - [3(9) – 6(4)] = - [27 – 24] = -3
Page 30
Matrices and its Applications: Albert Alipan Tayong
a33: +
3 4
3 4 2
2 7
2 7 5
6 9 8
a32 = [3(7) – 2(4)] = [21 – 8] = 13 Using the values above, we have the following:
Co-factors =
11
- 14
6
14
12
-11
-24
-3
13
Step 3: Find the Ad-joint. Note that the ad-joint of a square matrix is its transpose. Use the formula below to find the final answer.
Ad-joint of M =
A−1 3x3 =
M −1 =
1 41
1 đ??ˇđ?‘’đ?‘Ą
*
11
14
-24
-14
12
-3
6
-11
13
The rows of cofactors become the columns in the Adjoint of M. This is one of the applications of matrix transpose.
* Ad-joint of A 11
14
-24
-14
12
-3
6
-11
13
Page 31
=
11
14
−24
41
41
41
−14
12
−3
41
41
41
6
−11
13
41
41
41
Matrices and its Applications: Albert Alipan Tayong
14. Operations on Matrices using Excel Functions Excel is a very powerful tool used in solving various real-life equations and/or phenomena. In this section, we will be exploring the power of excel using matrices. In here, you will find that with the use of excel functions; operations on matrices can be performed in seconds.
Example 1: Find the sum.
A=
3 5 1
4 2 8
6 0 5
B=
2 -3 -3
10 4 -4
31 7 8
Solution: Input each value in the cell, separating both matrices.
A=
A+B=
3 5 1
4 2 8
6 0 5
5 2 -2
14 6 4
37 7 13
B=
2 -3 -3
10 4 -4
31 7 8
In order to get A + B, follow these steps: 1. Highlight 3x3 cells. That means 9 cells equivalent to 3 rows and 3 columns. 2. Press =. Then, highlight all elements of matrix A, and then press the + sign and finally highlight the elements of matrix B. 3. Press Ctrl Shift Enter simultaneously. To demonstrate the steps above, we have the following: =Highlight matrix A elements+Highlight matrix B elements and press Ctrl Shift Enter at the same time. Page 32
Matrices and its Applications: Albert Alipan Tayong
Example 2: Find C + B, A – C, and B + D.
A=
2 3 1 1 0 4 7 8 4
B=
1 9 -4 3 8 10
12 6 11
C=
1 1 4
8 15 2 11 19 1
7 D= 6
Steps: C + B 1. Highlight B10 to D12. This is because our matrices are 3x3. You can do this anywhere on the sheet. 2. Press the equal sign immediately after highlighting these cells. 3. Highlight all elements of matrix C. 4. Press the plus sign. 5. Highlight matrix B. 6. Press Ctrl Shift Enter all together. The values will then appear as shown above. Page 33
Matrices and its Applications: Albert Alipan Tayong
Steps: A - C 1. Highlight B15 to D17. This is because our matrices are 3x3. In here, we put it under C + B. Press the equal sign immediately after highlighting these cells. 2. Highlight all elements of matrix A. 3. Press the minus sign. 4. Highlight matrix C. 5. Press Ctrl Shift Enter all together. The values will then appear as shown above. Note: B + D is undefined so we will not be any more using excel for that. Page 34
Matrices and its Applications: Albert Alipan Tayong
Example 3: Find FC.
C=
3 7
1 4
2 8
F=
9 5
6 3
Steps: FC 1. Identify the dimension of each matrix and check whether FC is defined or undefined. After which identify the dimension of the answer (first and last values). 2. Highlight B12 to D13. This is because the desired matrix is 2x3. That means you need 2 rows and 3 columns. Bear in mind that you can put this anywhere the worksheet, depending on your choice. 3. Press =MMULT. Then, highlight matrix F. 4. Press comma and then highlight matrix C. 5. Press Ctrl Shift Enter all together. The values will then appear as shown above. Remark: MMULT stands for matrix multiplication. You need to follow the arrangement of letters. In here, we are asked to find FC. That means you need to highlight matrix F first then press comma (,) and finally highlight matrix C. Page 35
Matrices and its Applications: Albert Alipan Tayong
Example 4: Find P T .
P=
3
2
6
4
7
9
2
5
8
Steps: 1. Identify the dimension of the matrix. The dimension of the transpose is the opposite of the dimension of the original matrix. Interchange the rows and columns. The given matrix is 3x2, so its transpose is 2x3. 2. Highlight G8 to I9. This is because the desired matrix is 2x3. That means you need 2 rows and 3 columns. Bear in mind that you can put this anywhere the worksheet, depending on your choice 3. Press =TRANSPOSE and then highlight the given matrix. Press Ctrl Shift Enter all together. It then displays the answer.
Page 36
Matrices and its Applications: Albert Alipan Tayong
Example 5: Find the inverse of the following: 4
1
3
7
3
2
6
4
7
9
A=
P=
2
5
8
Steps: 1. Highlight corresponding rows and columns. If it is a 2x2 matrix, highlight 4 cells. If it is a 3x3 matrix, highlight 9 cells. If it is a 4x4 matrix, highlight 16 cells, and so on and so forth. 2. Press =MINVERSE (Highlight the given matrix.) Remark: All letters in the formula are not case sensitive. That means it can be small or big. All letters and symbols used in the formula must have no spaces. Use parentheses or ( ) always as your grouping symbols. Kindly have a look at the examples above and try the same using Excel functions. Page 37
Matrices and its Applications: Albert Alipan Tayong
15. Word Problems Involving Multiplication, Cramer’s Rule, and Inverse of Matrices Example 1: Cryptography is a science of information security. It is derived from the Greek word “kryptos”, meaning hidden. Encryption matrix or encoding matrix or invertible matrix (square matrix) – This is upon the discretion of the one encrypting the data. It is not transmitted. It is known by the receiver so that they can decrypt the message. It should not be sent with the data, otherwise anyone could grab the data and decode the information. By not having the decoding matrix, someone intercepting the message doesn’t know what size of matrix to use. The decryption matrix or decoding matrix or the inverse of encoding matrix (B −1 ) is to be calculated by the receiver in order to get XB −1 and decode the information. Find the password (all small letters) using the following matrices: Encrypted data:
X=
25
51
12
26
31
71
Receiving party has the encryption matrix (given below):
1
3
2
4
B=
1ST Step: Get the inverse of B. a11 = 1; a12 = 3
Page 38
Matrices and its Applications: Albert Alipan Tayong
a21 = 2; a22 = 4 Det B = 1(4) – 3(2) = -2
B
−1
=
B −1 =
B
−1
1
∗
Det
1 −2
=
a22
-a12
-a21
a11
4
∗
-3
-2
-2
1.5
1
-0.5
1
or
B
−1
=
-2
3/2
1
-1/2
2nd step: Find XB
XB −1 =
−1
:
25
51
12
26
31
71
*
-2
3/2
1
-1/2
(3 x 2)
(2 x 2)
Equal Dimension: 3 x 2 Page 39
Matrices and its Applications: Albert Alipan Tayong
XB −1 =
a11
a12
a21
a22
a31
a32
a11 = 25(-2) + 51(1) = 1 a12 = 25(3/2) + 51(-1/2) = 12 a11 = 12(-2) + 26(1) = 2 a12 = 12(3/2) + 26(-1/2) = 5 a11 = 31(-2) + 71(1) = 9 a12 = 31(3/2) + 71(-1/2) = 11
XB −1 =
1
12
2
5
9
11
Password: albeik Remark: a = 1, b = 2 … z = 26 Example 2: A Ticketing Agency had booked 2 flights for the next day. The first flight had 43 tickets bound for Thailand and 72 tickets bound for India, and totaled 21775 rials. The second flight had 54 tickets bound for Thailand and 46 tickets bound for India, and totaled 19350 rials. What were the price of each ticket bound for Thailand and each ticket bound for India? Use the Cramer's Rule to solve the problem. Let x = the price of each ticket bound for Thailand y = the price of each ticket bound for India
Page 40
Matrices and its Applications: Albert Alipan Tayong
Solution: System: 43x + 72y = 21775 54x + 46y = 19350
43
72
54
46
B=
A=
21775 19350
43
21775
54
19350
y= Det A = 43(46) - 72(54) = -1910
21775
Det y = 43(19350) -21775(54)
72
= -343800
x= 19350
y = Det y/Det A
46
y = -343800/-1910 y = 180
Det x = 21775(46) - 72(19350) = -391550
Hence, x =205 R.O. (Ticket price for Thailand)
x = Det x/Det A
y =180 R.O. (Ticket price for India)
x = -391550/-1910 = 205
Example 3: A University is purchasing projectors and speakers for their classrooms. The average cost is R.O. 250 and R.O. 56, respectively. If a total of 200 projectors and speakers is to be purchased with a budget of R.O. 29048, how many projectors and speakers will be required? Solve using the inverse of a matrix.
Page 41
Matrices and its Applications: Albert Alipan Tayong
Solution:
đ??´âˆ’1 =
Let x = projector
1 đ??ˇđ?‘’đ?‘Ą
đ?‘Ž22
*
−đ?‘Ž21
y = speaker
−đ?‘Ž12 đ?‘Ž11
System: x + y = 200 250x + 56y = 29048 Matrix:
1 A=
250
1
200
56
29048
a11 = 1;
a12 = 1;
Det A = 1(56) – 1(250) = -194
−1
A
−1
A
=
a22
1
∗
Det
1 = −194
-a12
-a21
a11
56
-1
=
* -250
1
X = A−1 * B Y
Page 42
−28
1
97
194
125
−1
97
194
a21 = 250;
a22 = 56
Matrices and its Applications: Albert Alipan Tayong
X
−28
1
97
194
=
*
Y
−28 97 125 97
1
(200) +
(200) +
200
194 −1 194
125
−1
97
194
29048
(29048) = 92
(29048) = 108
Hence, X
=
Y
92 108
So, x = 92 projectors
y= 108 speakers
Example 4: The entrance fee for a show in the Opera House was 20 Rials for Adults and 5 rials for children. During the show, the total amount collected was 40505 Omani Rials. If 2101 people watched the show, how many adults and how many children entered? Use a Cramer's Rule of matrix to solve the problem.
Let x = number of adults y = number of children
Page 43
Matrices and its Applications: Albert Alipan Tayong
Solution:
System: x + y = 2101 20x + 5y = 40505
A=
1
1
20
5
1
2101
20
40505
y=
B=
2101
Det y = 1(40505) -2101(20)
40505
Det y = -1515
y = Det y/Det A = -1515/-15 = 101 Det A = 1(5) - 20(1) = -15 Hence, x = 2000 Adults y =101 children 2101
1
40505
5
x=
Det x = 2101(5) - 1(40505) = -30000
x = Det x/Det A = -30000/-15 = 2000
Note that Cramer’s Rule or Inverse of a Matrix can either be used in solving systems of equations, which are of square matrix type. In reality, there are so many ways to solve systems of equations, but we only use these special methods because of its wide applications in computers, especially in Excel. Page 44
Matrices and its Applications: Albert Alipan Tayong
Example 5: A company had three orders with KM Furniture Ltd. The first order was for 5 beds and 4 cupboards, and totaled 600 rials. The second order was for 3 beds, 6 cupboards, and 9 chairs and totaled 585 rials. The third order was for 20 beds and 30 chairs, and totaled 2450 rials. What were the price of one bed, one cupboard, and one chair? Find the inverse of the matrix using Excel 2007- 2013 and use it to get the values of x, y, and z. Let x = price of one bed y = price of one cupboard z =price of one chair Solution: 5x + 4y = 600 3x + 6y + 9z = 585 20x + 30z = 2450
A=
5
4
0
3
6
9
20
0
30
600 B=
585 2450
Page 45
Matrices and its Applications: Albert Alipan Tayong
A−1
=
1/7
-2/21
1/35
1/14
5/42
-1/28
-2/21
4/63
After highlighting the values in matrix A, you need to press Ctrl Shift Enter simultaneously in order to get these values. For reference, kindly follow the steps shown above.
1/70
x = A−1
y
*B
z
A−1 * B
x=
y=
z=
1 7
=
(600) + (
1 14 −2 21
−2 21
(600) + (
(600) + (
1/7
-2/21
1/35
1/14
5/42
-1/28
-2/21
4/63
1/70
) (585) +
5 42 4 63
1 35
) (585) +
) (585) +
600 *
585 2450
(2450) = 100
−1 28 1 70
(2450) = 25
(2450) = 15
Hence, The bed is 100 rials each. The cupboard is 25 rials each. The chair is 15 rials each. Remark: The inverse of Matrix A can also be solved manually. Please refer to page 28 in order to review the steps. Page 46
Matrices and its Applications: Albert Alipan Tayong
To find the values of x, y, and z, you can also use the Excel function “=MMULT”, which stands for matrix multiplication. Using this function, you can multiply A−1 and Matrix B. The figure below shows you how to use the formula.
Follow these steps in order to get the values of the variables: 1. 2. 3. 4. 5.
Press =MMULT( Highlight A−1 as shown above. Press comma (,) Highlight Matrix B as shown above. Press Ctrl Shift Enter simultaneously.
The moment you press Ctrl Shift Enter simultaneously, the following values will then be displayed: x y z
100 25 15 Page 47
Matrices and its Applications: Albert Alipan Tayong
Example 6: Find the values of x, y, z, and p using an excel function. x + y + z + p = 14 2x + p = 9 3x + y + z = 13 2x + 5y + z + p = 28 1 2 3 2
A=
1 0 1 5
1 0 1 1
1 1 0 1
B=
14 9 13 28
Solution:
A=
−1
đ??´
=
1 2 3 2
1 0 1 5
1 0 1 1
1 1 0 1
-0.25
0.25
0.25
0.0000000000000000185
-0.188 0.9375 0.5
-0.063 -0.688 0.5
-0.063 0.3125 -0.5
0.25 -0.25 0
x y z p
2 3 4 5
B=
14 9 13 28
Steps: 1) Highlight 4x4 cells anywhere on the sheet. That means 4 rows and 4 columns. 2) Press “=MINVERSE(“ and then highlight Matrix A 3) Press Ctrl Shift Enter simultaneously. 4) Press =â€?MMULT (“and then highlight all elements of đ??´âˆ’1 . 5) Press comma and then highlight matrix B. And, finally press Ctrl Shift Enter simultaneously. Page 48
Matrices and its Applications: Albert Alipan Tayong
16. Exercise C
A. On the first day, Omar bought 3 apples and 5 bananas. On the second day, he bought 1 apple and 4 bananas. Granting that the prices of these fruits remain constant during a week, what is the price of each apple and each banana? The figures and total prices are given below. Use a Cramer’s rule to solve the problem.
1. + +
=
1.110 OMR
+
=
0.720 OMR
2.
B. Al Juman School had 2 orders from Al Manara Bookshop. The first order was for 203 English books and 150 Mathematics books, and totaled 2418 OMR. The second order was for 100 English books and 30 Mathematics books, and totaled 840 OMR. What was the price of each English book and each Mathematics book? Use a matrix inverse to solve the problem. C. 10000 Dirhams is divided between two accounts, one paying 6.5% interest and the other paying 7% interest. At the end of one interest period, the interest earned by the 7% account exceeds the interest earned by the 6.5% account by 214 Dirhams. How much was invested in each account? Use the Cramer's rule for matrix to solve the problem. Let .065x = interest of first account .07y = interest of second account
Page 49
Matrices and its Applications: Albert Alipan Tayong
D. Find the value of x, y, and z using excel functions. 3x + 2y + z = 23 2x - 5y + 10 z = -1 x + y + 4z = 21 E. In a festival, the entrance fee for adult is 4 rials and the entrance fee for children is 2 rials. During a particular day, 3000 people entered and 9600 OMR was collected. How many children and how many adults entered? Solve the problem using a matrix inverse. F. Solve by Cramer's rule: 3x - 2y = 7 2x + 4y = -6
G. Find the value of x and y using a matrix inverse: 5x + y = 16 x + 6y = 9 H. Solve the following systems of equations using excel functions: 2x + 3b + s + 2m = 59 2x + m = 13 4x + b + s = 19 8x + 3b + 5s + m = 74
a + 9b + 3c + 5d + e = 49 3a + b = 7 6a + 2b + 3c + d = 32 7a + 10b + 5c = 49 a + b + e = 11
Page 50
Matrices and its Applications: Albert Alipan Tayong
17. Answers: Page 15, Exercise A
1)
D+F=
5
13
11
5
47
10
1
16
29
2) M – Y is undefined because M is a 3x3 matrix and Y is a 3x1 matrix.
N+M=
B+N=
3
10
28
8
-2
17
12
11
15
8
15
32
4
0
15
11
19
15
3)
6 A+A=
5 1 4
B+C=
14
6
42
8
5 9
7
B-A=
−9
−33
40
40
5
7
−9
16
27
0
4
8
6 A-B=
−85 11
9
33
−40
40
5
7
69
9
11
8
Page 51
4
-4
85 11
Matrices and its Applications: Albert Alipan Tayong
4)
D-F=
3
-5
5
5
-41
-12
3
20
-5
-4
-4
-8
-5
-3
1
-2
-18
-12
5)
F=
18. Answers: Page 24, Exercise B 1) Comp.
A = Office 1 B = Office 2 C = Office 3
Office Mngr Acct A 1 2 B 1 2 C 1 2
HR 1 1 1
Arch 2 3 3
Eng 3 4 4
Adm Des Sec 2 4 2 0 6 3 0 6 3
Office Mngr Acct A+B+C 3 6 Total 83
HR 3
Arch 8
Eng 11
Adm 2
Des 16
Cl 2 3 3
Mktg Sales 0 0 4 5 4 5
Sec Cl Mktg Sales 8 8 8 10
Hence, a) Managers = 3, Accountants = 6, HR Officers = 3, Architects = 8, Engineers = 11, Admin Officers = 2, Designers = 16, Secretaries = 8, Cleaners = 8, Marketing Officers = 8, Salesmen = 10 b) Total number of employees = 83 Page 52
Matrices and its Applications: Albert Alipan Tayong
2)
BA=
KT =
133 42 34
132.5 56 37
8 1/3
2/5 1/7
6 8
3 2 1/4 6 3/4
3/4 C
3)
136.5 37 43
2+5
12 15
- 3/8 30 46 1/2
6+1
A+B=
5+2
1+6
B+A= 3+7 7
4+8
7+ 3
7
A+B=
8+4
7
7
10
12
B+A= 10
12
Therefore, A+ B = B + A Page 53
Matrices and its Applications: Albert Alipan Tayong
C=
1/4 2
5 1/2
2.20 4
D=
5 2
10 3/4
8 14
C-D=
-4.75 0
-5 -0.25
-5.8 -10
D-C=
4.75 0
5 0.25
5.8 10
Therefore, C – D ≠ D – C.
19. Answers: Page 49, Exercise C A) 3x + 5y = 1.110 x + 4y = 0.720 x = 0.12 or 120 baize (price of each apple) y = 0.15 or 150 baize (price of each banana) B) Eq. 1 Eq. 2
Inverse:
203 100
150 30
= =
2418 840
-0.00337 0.016835 0.011223 -0.02278
x=
y=
6
Price of an English book
8
Price of a Mathematics book
Page 54
Matrices and its Applications: Albert Alipan Tayong
C) Eq. 1 Eq. 2
1 0.065
1 -0.07
= =
10000 -214
x = 3600 Dirhams, y = 6400 Dirhams D) Eq. 1 Eq. 2 Eq. 3
Inverse:
3 2 1
2 -5 1
1 10 4
0.379747 0.088608 -0.31646 -0.025316 -0.13924 0.35443 -0.088608 0.012658 0.240506
x= y= z=
2 7 3
Step 1: MINVERSE FUNCTION Step 2: MMULT FUNCTION E) x = adult = 1800 y = children = 1200 F) x = 1, y = -2
G) x = 3, y = 1
H) x = 1, b = 10, s = 5, m = 11 a = 2, b = 1, c = 5, d = 3, e = 8
Page 55
= = =
23 -1 21
Matrices and its Applications: Albert Alipan Tayong
20. Bibliography: Books: 1. Khanna, Qasi Z VK, (1991), Mathematics in Commerce & Economics. 2nd Edition. Vikas Publishing House PVT LTD. 2. Seymour, L., Marc, L., (2006), Schaum’s Outline of Linear Algebra, 3rd Edition, McGraw-Hill. 3. Susanna, E., (2003), Discrete Mathematics with Applications, 3rd Edition. World Wide Web Page: 1. Pearson India Education Services Pvt. Ltd (Formerly TutorVista Global Private Limited), (2015) Application of Matrices. [ONLINE] available at: http://www.edurite.com/kbase/application-of-matricesin-real-life (Accessed 23 August 2015) 2. (2015) Photos of bananas. [ONLINE] available at: http://www.bing.com/search?q=photos+of+bananas&src=IETopResult&FORM=IE11TR&conversationid= (Accessed 23 March 2015) 3. Using MS Excel in Matrix Multiplication (2015) [ONLINE] available at: http://www.math.iupui.edu/~momran/m118/matrices2.pdf (Accessed 12 Mar 2015) 4. Fatima, et al (2015) Matrix Inverse Function. [ONLINE] available at: http://www.addictivetips.com/microsoft-office/excel-2010-matrixinverse-function-minverse/ (Accessed 25 April 2015) 5. Chen 3600- Computer- Aided Chemical Engineering, Chemical Engineering Department Notes 4, EWE: “Engineering With Excel” Larsen (2015), Matrix Operations in Excel [ONLINE] available at: http://www.eng.auburn.edu/~clemept/CEANALYSIS_SPRING2011/ matrixoperations_notes.pdf (Accessed 28 February 2015) 6. (2015) Cryptography [ONLINE] available at: http://aix1.uottawa.ca/~jkhoury/cryptography.htm (Accessed 5 May 2015) 7. Ikenaga, Bruce (2012) Word Problems Involving Systems of Linear Equations [ONLINE] available at: http://www.millersville.edu/~bikenaga/basic-algebra/systems-wordproblems/systems-word-problems.pdf (Accessed 20 June 2015) 8. S. Taylor- Algebra Lab (2003-2015) Operations with matrices [ONLINE] available at: http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix _operations.xml (Accessed 5 March 2015) Page 56