FlipWorkbookDUM20132

DUM21032
ENGINEERING
MATHEMATICS 2

Workbook & Tutorial

UNIT 1 MATRICES 1-44

UNIT 2 VECTORS 45-63

UNIT 3 DIFFERENTIATION 64-86

UNIT 4 INTEGRATION 87-135

UNIT 1 DUM20132 ENGINEERING MATHEMATICS 2
UNIT 1: MATRICES

INTRODUCTION

The matrix has a long history of application in solving linear equations. They were known as arrays until
the 1800’s. The term “matrix” (Latin for “womb”, derived from mater—mother) was coined by James Joseph
Sylvester in 1850, who understood a matrix as an object giving rise to a number of determinants today called
minors, that is to say, determinants of smaller matrices that are derived from the original one by removing
columns and rows. An English mathematician named Cullis was the first to use modern bracket notation for
matrices in 1913 and he simultaneously demonstrated the first significant use of the notation A=ai,j to represent
a matrix where ai,j refers to the element found in the ith row and the jth column. Matrices can be used to
compactly write and work with multiple linear equations, referred to as a system of linear equations,
simultaneously. Matrices and matrix multiplication reveal their essential features when related to linear
transformations, also known as linear maps. Matrices are used a lot in daily life.

CONTENT

1.1 DEFINITION
1.2 OPERATION OF MATRICES
1.3 SOLVING SYSTEM OF LINEAR EQUATIONS

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UNIT LEARNING After completing the unit, students should be able to:
OUTCOMES 1.1.1 Identify matrix and its properties.
1.1.2 Identify the various types of matrices
1.1 DEFINITION 1.1.3
1.1.4 Convert the angle in degree to radian and vice versa.

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EXERCISE 1.1

1. Complete the table below. Number of rows Number of columns Order
Matrix 3

 2 − 5 3 rows
3 1
 −1 6  2 3x2

(read as “ 3 by 2” matrix)

2 rows

a)  52

 3 0 5
b)  2 4 7

 −1 0 2

c)  9 7 −42 
1 0

2. State the elements given in each of the following matrices.

A = (− 8 2 − 3 1) B = 150.6.53 40.2 − 1.25 
250 2.12
a12 =
a13 = b11 =
b21 =
b23 =
b22 =

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3. Find the values of p and q for each of the following.

 p  =  4−p   p + 1  = 7 − p 
2 -q 3 q

 p− 1 =  5 − p   − 6 12   − 6 12 
6 3 − q 1 + 5 p 5  =  10 5
 3 2q   3 8 + q 

4. Given A =  3 1  , B =  2 1  and C  3 
2 5 0 −6 =  2  . Find

 −1

a) AT b) BT c) C T

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TUTORIAL 1.1

1. Complete the following table

Matrix [1 −3 5] [24] [75] [14 −23] 1 13
[12 22]
−7 −6

No of rows

No of columns

Order of matrix

2. Determine the order of each of the following matrices.

a) [13 −2 −35] 30
4 b) [9 −5]

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3. Determine type of matrices below whether it is a column matrix, row matrix, square matrix or identity

matrix

−4 b) [32 −54]
a) [ 2 ]

5

c) [1 −2 5] 100
d) [0 1 0]

001

e) [7 8] f) [51]

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5 −3
4. Given matrix A = [ 0 1 ] .

−1 9
a) Identify

i) the order of matrix A.

ii) the elements of a11 , a12 and a32.

b) Find the transpose of matrix A

5. Given A = [43 2x] and B [3y 25]. Find the values of x and y if A = B.

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UNIT LEARNING After completing the unit, students should be able to:
OUTCOMES 1.2.1 Operation of matrices.

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EXERCISE 1.2 b)  6  +  −7 
1. Calculate. 8 2

a) (9 − 5) + (10 − 4)

c)  3 − 13 +  5 6   3 − 5  1 − 2
5 − −2 9 d)  − 4 2  −  − 1 4

 6 − 1  3 − 6 

e)  2 − 13  +  1 6   2 − 5  4 − 2
4 −3 −2 3 f)  − 3 2  −  − 1 − 5

 7 − 1  2 − 3

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2. Given that P = 5 2 Q = − 5 1 and R = 0 − 7
− 1 4 , − 3 6 2 − 9 . Express each of the following

as a single matrix.

a) P − Q + 2R b) P
5

c) 2Q + RT d) (3P + Q)T − R

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3. Find the values of x and y for each of the following matrix equations.

a) (x 4 y + 2) + (− 2 − 5 6) = (0 −1 12) b)  4 − 8 y  −  1 −x3 =  3 −2 
1 2− −4 5 7

c) 5(2x − y) − 1 (4x 12) = (− 8 − 2y) d)  2   y   1 
− 2 x + 2 +  − 7 =  −13
2  − 2   4   8 

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4. Find the following products (where possible)

a)  2 13 2 3 04  b) 13(2 4 6)
1 −1 1
 5 

(6 5 3) 1  d) (− 2 5) 4 3 
2  −4 −5
c)

 − 2

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 5 0  −9 84  f)  2 20.5  1 10 
−2 5 1 −6 0
e)

g)  2 3 04  2 13  1 − 3 612 
−1 1 1 h)  5 1 215

 − 2 3 118

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TUTORIAL 1.2

1. Given X = [−01 63], Y = [−51 23] and Z = [41 −35]. Find:

a) X + Y b) Y+Z

c) Y − Z d) Z − X

e) Y + 2X − 3Z f) 3Z − X + 2Y

2. Given A = [62 −1 −14]and B = [50 3 −97], find:
3 −1

a) A − B b) A+B

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c) 3A + 2B − A DUM20132 ENGINEERING MATHEMATICS 2
UNIT 1: MATRICES

d) B − 2A

3. Find the value of x and y each of the following matrices.
a) [8x] = [27y] + [42]

b) [x −6] = [3x y] − [5 2y]

c) [3y 24x] + [4x −3xx] = [67 −5x1]

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4. Find matrix Q if[−62] + Q = [47]

5. Find the product each of the following: 3
a) [7 1] [−25] b) [6 −2 4] [ 1 ]

−5

c) [−23] [8 4] −9
d) [ 2 ] [7 −1 0]

3

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e) [12 34] [−32] f) 1 2 [−21]
[0 −3]
5 4

g) [5 2] [−12 3 −51] h) [−21 8 36] 3 5
6 0 [1 −2]
3 −1

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UNIT LEARNING After completing the unit, students should be able to:
OUTCOMES 1.3 Solve systems of linear equations up to three variable by using:

1.3.1 Matrix Inversion Method for 2 x 2 matrix
1.3.2 Cramer’s rule for 2 x 2 matrix
1.3.3 Matrix Inversion Method for 3 x 3 matrix
1.3.4 Cramer’s rule for 3 x 3 matrix

1.3 SOLVING SYSTEMS OF LINEAR EQUATION
1.1.1

1.1.2 Convert the angle in degree to radian and vice versa.

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EXERCISE 1.3.1

1. Find the inverse of the following matrix

a)  32 − 34  b)  1 −2 
− −3 7

c)  2 41  d)  − 3 −− 41 
6 − 2

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2. Solve the following simultaneous equation by using matrix inversion method
a) 2x – 3y = 11
x + 2y = -5

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b) 3p – q = 3
p – 3q = 5

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c) 4x + y = 10
2x – 3y = 12

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EXERCISE 1.3.2
1. Use Cramer’s Rule to solve the following simultaneous equations :

a. 3x + 5y = 7
4x – 3y = 19

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b. 4x + y – 10 = 0
2x – 3y – 12 = 0

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EXERCISE 1.3.3  2 − 3 1 
1. Find the inverse of the following matrix b)  − 3 − 1 5 

 − 2 3 0  2 4 − 5
a)  4 − 1 5

 2 4 6

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2. Solve the following simultaneous equation by using matrix inversion method
a) 2x – y + 3z = 2
x + 3y – z = 11
2x – 2y + 5z = 3

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b) 2x – y – 3z = -8
x + 3y + 2z = 3
5x + 2y + z = 9

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EXERCISE 1.3.4
1. Use Cramer’s Rule to solve the following simultaneous equations :

a. x + y + z = 4
2x – 3y + 4z = 33
3x – 2y -2z = 2

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b. x + 2y + 2z - 4 = 0
3x – y + 4z – 25 = 0
3x + 2y – z = -4

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TUTORIAL 1.3

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UNIT 2 VECTORS

INTRODUCTION

The word ’vector’ comes from Latin and means’ carrier’. As the name suggested, quantities with direction and magnitude are
vectors. For example, a force has a magnitude (small or large according to scale) and a direction in which it acts. Another
example is velocity. On the other hand, quantities such as length, area, temperature are called scalar. These quantities which
can be specified by a single number for example, 4 mm and 20 U and have no directions.

CONTENT
2.1 DEFINITION
2.2 VECTOR ALGEBRA
2.3 COMPONENTS OF VECTOR
2.4 SCALAR PRODUCT
2.5 VECTOR PRODUCT

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UNIT 2: VECTOR

UNIT LEARNING After completing the unit, students should be able to:
OUTCOMES 2.1.1 Identify the notation of vector
2.1.2 state the definition of scalar and vector, zero vector, unit vector,
negative vector and equal vector
1.1.1 Convert the angle in degree to radian and vice versa.

2.1 DEFINITION Definition

Concept of Vector Physical quantities can be divided into two main
Scalar and vector groups,scalar quantities and vector quantities.
quantities a) A vector quantity is one which is specified by its

magnitude and direction, e.g. displacement,
velocity,acceleration, force, weight, etc.
b) A scalar quantity is one which is specified by its
magnitude only e.g. length, area, volume, mass,
time,distance, speed, temperature, etc.

Vector representation Vector quantity can be represented graphically by a line,
and notation of vector drawn so that:

a) the length of the line denotes the magnitude of
thequantity,

b) the direction of the line denotes the direction in
whichthe vector quantity acts.

Vectors are denoted by:
a) bold type eg. a , AB

b) two capital letters with an arrow above them to
→

describe the sense of direction eg. AB where A is the
starting point and B the end point of the vector

→→

c) letters with an arrow above eg. a , A

d) underlined letter eg. a

B

→→
a or AB

A

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Modulus DUM20132 ENGINEERING MATHEMATICS 2
UNIT 2: VECTOR
Zero vector
Unit vector →→
Negative vector
The modulus of the vector a is the magnitude of a , that is the
Equal vector
Position vector →→

length of the line representing a . The modulus of a is written

→

as a .

A zero vector, or null vector is a vector which has
zero magnitude and an arbitrary direction. It is denoted
by 0 .

A unit vector is a vector which has a magnitude of 1 unit.
Special unit vectors are the vector i , j and k , which
are vectors in the direction of positive x-axis, y-axis and
z-axis respectively.

A negative vector has the same magnitude but in the
→→

opposite direction. Hence − AB = BA

BB
→
AB →

BA

AA

Two vectors are equal if and only if they have the
samemagnitude and the same direction.
If O is the origin, the vector represented by the directed

→
line OA is called the position vector of A.

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UNIT LEARNING DUM20132 ENGINEERING MATHEMATICS 2
OUTCOMES UNIT 2: VECTOR

After completing the unit, students should be able to:
2.2.1 perform the standard operation with vector including
• Addition
• multiplication

2.2 VECTOR ALGEBRA

VECTOR ALGEBRA CONCEPT
ADDITION OF
VECTORS Triangle Law

The addition of vectors is defined as follows.

C

a+b

b

Aa B → B

If the sides AB and BC of triangle ABC represent the vectors

→→

a and b , the third side AC represent the vector sum, or

→→ →→

resultant of a and b and is denoted by a + b .

→→ →
AB + BC = AC

Parallelogram law

If the sides OA , OB of a parallelogram OACB represent
vectors a and b respectively , the diagonal OC represents
the sum of a + b .

A C
B
a a+b
O b

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