Advanced Algebra

Inequalities

Trichotomy Property of Order

If a and b are real numbers, exactly one of the following three statements are true

a < b b < a a =b

Properties of <

If a, b and c are real numbers and

i. If a < b, then a + c < b + c (Addition Property)

ii. If a < b, then a – c < b – c (Subtraction Property)

iii. If a < b, and c > 0, the ac < bc(Multiplication Property)

iv. If a < b, and c < 0, the ac > bc(Multiplication Property)

Example 1. Find the solution set of

the inequality3x – 8 < 7

Solution: 3x – 8 < 7

3x – 8 + 8 < 7 + 8

3x < 15

(1/3) 3x < 15 (1/3)

x < 5

Exaple 2. Find the Solution set of

the inequality

Solution:

Progression

Arithmetic Progression

A sequence of numbers in which the

difference of any two adjacent terms is

constant.

Example: 4,7,10,13,16,… (common

difference = 3)

ARITHMETIC MEAN: The terms

between any two given terms of

arithmetic progression are called

arithmetic mean.

Elements:

a1 = first term

an = nth term

am=any term before an

d= common difference d= a2-a1=a3-

a2=a6-a5, etc

S = Sum of all the terms

an = a1 + (n-1)d or an= am + (n-m)d

S = n/2 (a1 + an) or S = n/2 [ 2a1 + (n-1)

d]

Geometric Progression

A sequence of numbers in which any

two adjacent terms has a common

ratio

Example: 2, 6, 18,54, … ( common

ratio, r = 3)

Geometric Mean: The terms between

any two given terms of a geometric

progression is called geometric means

between the given terms.

an= a1rn-1 or an=amrn-m

Common ratio, r = a2/a1= a5/a4=…

S = a1(rn-1)/ r-1 when r > 1

S = a1( 1-rn)/1-r when r < 1

Sum of Infinite Geometic Progression

= a1/ 1-r

Harmonic Progression

A sequence of numbers in which their

reciprocals forms an arithmetic

progression

Example : ½, ¼, 1/6, 1/8, …

HARMONIC MEAN: The terms of a

harmonic progression between any

two given terms are called harmonic

means

Permutation

Permutation refers to arrangement of

objects in a definite order.

Permutations of n things in a circle

( Cyclical Permutation)

P = (n-1)!

The permutation of n different things

taken r at a time is :

P (n,r) = n! / (n-r)! and P (n,n) = n!

Note 0! =1

Example: How many permutations can

be made out of the letters in the word

DIEGO taken 3 at a time?

Solution: n=5, r=3

P (5,3) = 5!/ (5-3)! = 60 ways

The permutation of n things of which q

are alike, r are alike, and so on is:

P = n!/ q! r! …

Example: How many permutations can

be made out of the letters in the word

GILLESANIA?

Solutions: n = 10, (2-I, 2-L, 2-A)

P = 10! / 2! 2! 2! = 453,600 ways

Combination

Combination refers to a collection of

objects without regard to sequence or

order of arrangement.

Combination of n things taken r at a

time:

C(n,r) = P (n, r)/ r! = n! / (n-r)! r! and C

(n,n) = 1

Example: How many ways can you

draw 3 QUEENs and 2 KINGs from a

deck of 52 cards?

Solution: A deck of 52 cards has 4

QUEENs and 4 KINGs, thus:

C= C (4,3) x (4,2) = 24 ways

Combination of n things taken 1,2,3 …

n at a time:

C = 2n-1

Example: How many ways can you

invite any one or more of your five

friends to your birthday party?

Solution : C = 25-1 = 31 ways

Matrices and Determinants

Matrix

A matrix is a rectangular collection of

variables or scalars contained within a

set of square [ ] or round ( ) brackets.

A matrix consists of m rows and n

columns.

Classification of matrices

Square Matrix

A matrix whose number of rows m is

equal to the number of columns n.

Diagonal Matrix

A diagonal matrix is a square matrix

with all zero values except for the aij

value for all i=j

Identity Matrix

An identity matrix is a diagonal matrix

with all non- zero entries equal to 1.

Scalar Matrix

A scalar matrix is a diagonal matrix with

all non- zero entries equal to some

other constant.

Triangular Matrix

A triangular matrix has zeros in all

positions above or below the diagonal

MATRIX is an array numbers

Adding two matrices: Subtracting two matrices:

Multiplying matrices:

Identity matrix:

INVERSE OF A MATRIX:

Example:Formula:

Check if the answer is correct:

Inverse of a Matrix

using Elementary Row

Operations:

Example:

• The "Elementary Row

Operations" are simple things

like adding rows, multiplying

and swapping

Inverse of a Matrix

using Minors, Cofactors and

Adjugate: Example:

Step 1: Matrix of Minors

Calculation for the whole matrix:

Step 2: Matrix of Cofactors

Step 3: Adjugate (also called Adjoint)

Step 4: Multiply by 1/Determinant

Solving system linear equations

using matrices:

Solve:

• 𝑥 + 𝑦 + 𝑧 = 6• 2𝑦 + 5𝑧 = −4• 2𝑥 + 5𝑦 − 𝑧 = 27

𝑥 + 𝑦 + 𝑧 = 62𝑦 + 5𝑧 = −4

2𝑥 + 5𝑦 − 𝑧 = 27

1 1 1 = 60 2 5 = −42 5 − 1 = 27

A X = B

X = 𝐴−1 𝐵

𝐴−1=

The determinant of a matrix is a

special number that can be calculated

from a square matrix.

Example:

==

Example:

For a 2×2 Matrix:

For a 3×3 Matrix:

For a 4×4 Matrix:

=

=

=

Example:

=

Determinants of order two:

𝑎1𝑥 + 𝑏1𝑦 = 𝑘1𝑎2𝑥 + 𝑏2𝑦 = 𝑘2

𝑎1𝑏2𝑥 + 𝑏1𝑏2𝑦 = 𝑘1𝑏2𝑎2𝑏1𝑥 + 𝑏1𝑏2𝑦 = 𝑘2𝑏1

(𝑎1𝑏2𝑥 − 𝑎2𝑏2)𝑥 = 𝑘1𝑏2 − 𝑘2𝑏1

𝑥= 𝑘1𝑏2−𝑘2𝑏1

𝑎1𝑏2−𝑎2𝑏2

𝑦= 𝑘2𝑎1−𝑘1𝑎2

𝑎1𝑏2−𝑎2𝑏2

Derivation of formula:

=

𝑘1 𝑏1𝑘2 𝑏2𝑎1 𝑏1𝑎2 𝑏2

=

𝑎1 𝑘1𝑎2 𝑘2𝑎1 𝑏1𝑎2 𝑏2

Example:

2𝑥 − 3𝑦 = 16,5𝑥 + 2𝑦 = 2

𝑥 =

16 −32 22 −35 2

=16 ∙ 2 − 2 ∙ (−3)

2 ∙ 2 − 5 ∙ (−3)=38

19= 𝟐

𝑦 =

2 165 22 −35 2

=2 ∙ 2 − 16 ∙ 5

2 ∙ 2 − 5 ∙ (−3)=−76

19= −𝟒

Determinants of order three:

𝑥 =

𝑘1 𝑏1 𝑐1𝑘2 𝑏2 𝑐2𝑘3 𝑏3 𝑐3

𝑎1 𝑏1 𝑐1𝑎2 𝑏2 𝑐2𝑎3 𝑏3 𝑐3

𝑦 =

𝑎1 𝑘1 𝑐1𝑎2 𝑘2 𝑐2𝑎3 𝑘3 𝑐3

𝑎1 𝑏1 𝑐1𝑎2 𝑏2 𝑐2𝑎3 𝑏3 𝑐3

𝑧 =

𝑎1 𝑏1 𝑘1𝑎2 𝑏2 𝑘2𝑎3 𝑏3 𝑘3

𝑎1 𝑏1 𝑐1𝑎2 𝑏2 𝑐2𝑎3 𝑏3 𝑐3

Example:

Solve by determinants:𝑥 + 𝑦 + 𝑧 = 62𝑦 + 5𝑧 = −4

2𝑥 + 5𝑦 − 𝑧 = 27

𝑥 =

6 1 1−4 2 527 5 −11 1 10 2 52 5 −1

= 𝟓

𝑦 =

1 6 10 −4 52 27 −11 1 10 2 52 5 −1

= 𝟑

𝑧 =

1 1 60 2 −42 5 271 1 10 2 52 5 −1

= −𝟐