Advanced algebra

1. Advanced Algebra

2. 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)

3. Example 1. Find the solution set of
the inequality
3x – 8 < 7
Solution: 3x – 8 < 7
3x – 8 + 8 < 7 + 8
3x < 15
(1/3) 3x < 15 (1/3)
x < 5

4. Exaple 2. Find the Solution set of
the inequality
Solution:

5. Progression

6. 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.

7.  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]

8. 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.

9.  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

10. 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

11. Permutation
Permutation refers to arrangement of
objects in a definite order.
 Permutations of n things in a circle
( Cyclical Permutation)
P = (n-1)!

12.  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

13.  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

14. Combination
 Combination refers to a collection of
objects without regard to sequence or
order of arrangement.

15.  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

16.  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

17. Matrices and Determinants

18. 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.

19. 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

20.  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.

21.  Triangular Matrix
A triangular matrix has zeros in all
positions above or below the diagonal

22. MATRIX is an array numbers
Adding two matrices: Subtracting two matrices:

23. Multiplying matrices:

24. Identity matrix:

25. INVERSE OF A MATRIX:
Example:Formula:
Check if the answer is correct:

26. Inverse of a Matrix
using Elementary Row
Operations:
Example:
• The "Elementary Row
Operations" are simple things
like adding rows, multiplying
and swapping

27. 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

28. Solving system linear equations
using matrices:
Solve:
• 𝑥 + 𝑦 + 𝑧 = 6
• 2𝑦 + 5𝑧 = −4
• 2𝑥 + 5𝑦 − 𝑧 = 27
𝑥 + 𝑦 + 𝑧 = 6
2𝑦 + 5𝑧 = −4
2𝑥 + 5𝑦 − 𝑧 = 27
1 1 1 = 6
0 2 5 = −4
2 5 − 1 = 27
A X = B
X = 𝐴−1 𝐵
𝐴−1
=

29. 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:

30. For a 4×4 Matrix:
=
=
=
Example:
=

31. Determinants of order two:
𝑎1 𝑥 + 𝑏1 𝑦 = 𝑘1
𝑎2 𝑥 + 𝑏2 𝑦 = 𝑘2
𝑎1 𝑏2 𝑥 + 𝑏1 𝑏2 𝑦 = 𝑘1 𝑏2
𝑎2 𝑏1 𝑥 + 𝑏1 𝑏2 𝑦 = 𝑘2 𝑏1
(𝑎1 𝑏2 𝑥 − 𝑎2 𝑏2)𝑥 = 𝑘1 𝑏2 − 𝑘2 𝑏1
𝑥=
𝑘1 𝑏2−𝑘2 𝑏1
𝑎1 𝑏2−𝑎2 𝑏2
𝑦=
𝑘2 𝑎1−𝑘1 𝑎2
𝑎1 𝑏2−𝑎2 𝑏2
Derivation of formula:
=
𝑘1 𝑏1
𝑘2 𝑏2
𝑎1 𝑏1
𝑎2 𝑏2
=
𝑎1 𝑘1
𝑎2 𝑘2
𝑎1 𝑏1
𝑎2 𝑏2
Example:
2𝑥 − 3𝑦 = 16,
5𝑥 + 2𝑦 = 2
𝑥 =
16 −3
2 2
2 −3
5 2
=
16 ∙ 2 − 2 ∙ (−3)
2 ∙ 2 − 5 ∙ (−3)
=
38
19
= 𝟐
𝑦 =
2 16
5 2
2 −3
5 2
=
2 ∙ 2 − 16 ∙ 5
2 ∙ 2 − 5 ∙ (−3)
=
−76
19
= −𝟒

32. Determinants of order three:
𝑥 =
𝑘1 𝑏1 𝑐1
𝑘2 𝑏2 𝑐2
𝑘3 𝑏3 𝑐3
𝑎1 𝑏1 𝑐1
𝑎2 𝑏2 𝑐2
𝑎3 𝑏3 𝑐3
𝑦 =
𝑎1 𝑘1 𝑐1
𝑎2 𝑘2 𝑐2
𝑎3 𝑘3 𝑐3
𝑎1 𝑏1 𝑐1
𝑎2 𝑏2 𝑐2
𝑎3 𝑏3 𝑐3
𝑧 =
𝑎1 𝑏1 𝑘1
𝑎2 𝑏2 𝑘2
𝑎3 𝑏3 𝑘3
𝑎1 𝑏1 𝑐1
𝑎2 𝑏2 𝑐2
𝑎3 𝑏3 𝑐3
Example:
Solve by determinants:
𝑥 + 𝑦 + 𝑧 = 6
2𝑦 + 5𝑧 = −4
2𝑥 + 5𝑦 − 𝑧 = 27
𝑥 =
6 1 1
−4 2 5
27 5 −1
1 1 1
0 2 5
2 5 −1
= 𝟓
𝑦 =
1 6 1
0 −4 5
2 27 −1
1 1 1
0 2 5
2 5 −1
= 𝟑
𝑧 =
1 1 6
0 2 −4
2 5 27
1 1 1
0 2 5
2 5 −1
= −𝟐