Co factors of a matrix and Cramer's rule - Method of solving

Co-factor of a matrix

There are 2 methods, which are simple, to find a co factor matrix:

1.Long method

2.Short method

The long method

Assume that we are considering a 2*2 matrix named A.

The above is the method to find the co factors of a 2*2 matrix. We can use the same method to find the cofactor matrix for a 3*3 matrix too. 

The short cut method

For a better understanding we consider a 3*3 matrix and we name it A.

Now we have to find the determinants of each aligned semi matrices and the result would be the co cofactor matrix.

Cramer's rule

We can find the values of the variables using the Cramer's rule, where we'll have to use the linear equations to find out the answer;

For a better understanding of the method let’s consider a 2*2 matrix; and hence 2 linear equations would be given:

HCF and LCM -theory; meanings, methods and formulae

First we need to understand the meanings of factors and multiples.
Factors and multiples;


When a number a divides another number b exactly, we say that a is a factor of b. In this case, b is known as the multiple of a.

Highest Common Factor(HCF) or Greatest Common Measure(GCM) or Greatest Common Divider(GCD);
The HCF of two or more than two numbers is the greatest number that divides each of them exactly.
2 methods to find the HCF of a set of numbers:
1.Factorisation method: Express each one of the given numbers as the product of prime factors. The product of least powers of common prime factors given HCF. 
2.Division method: Assume that we have to calculate the HCF of two given numbers. Now, we divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is the required HCF.

Finding the HCF of more than 2 numbers: Sometimes, we'll have to find the HCF of 3 numbers. Then, HCF of [(HCF of any two) and (the third number)] gives the HCF of three given numbers.
Similarly, the HCF of more than 3 numbers can also be obtained using the above mentioned method.

Least Common Multiple(LCM);
The least number which is exactly divisible by each one of the given numbers is called their LCM.
2 methods;
1.Factorisation method of finding the LCM: Resolve each one of the given numbers into a product of prime factors. Then, LCM, is the product of highest powers of all the factors.
2.Common Division method(Short cut method) of finding LCM: Arrange the numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required LCM of the given numbers.

The product of 2 numbers is the product of their HCF and LCM.

2 numbers are said to be co primes if their HCF is 1.

HCF and LCM of fractions are represented by;
1.HCF = HCF of numerators / LCM of denominators
2.LCM = LCM of numerators / HCF of denominators
Decimal fractions;
In given numbers, make the same number of decimal places by annexing(add) zeros in some numbers, if necessary. Considering these numbers without decimal point, find HCF or LCM as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.

Comparison of fractions;
Find the LCM of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with LCM as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

Time series - meaning, components and examples, methods

Time series is the arrangement of data according to time of occurrence.

There are a few factors that are responsible for bringing about changes in a time series data. They are known as the components of time series.
The components:
1.Secular movements
2.Seasonal movements
3.Cyclical movements
4.Irregular movements

Secular movements are also known as secular trends. They are the long term variations. These variations take a long time to happen.
Example: 
Expansion of technology and increase in the quality of commodities.

Seasonal movements or seasonal variations are the short term variations. The variations takes time periods like the seasons(generally, 3 to 6 months to be approximate).
Example:
Increase in the sale of umbrellas during the rainy season.

Cyclical movements are like business cycles. The variations rise and fall. They follow a pattern of prosperity, decline, recession and a boom.
Example:
An up and down of the sales of a business due to a few reasons.

Irregular movements are also known as erratic variations. As the name suggests, the variations happen irregularly. Generally, the causes are natural disasters. 
Example:
Shutting down of a small restaurant due to it's destruction from a tsunami.

3 methods to calculate the time series:
1.Semi average method
2.Moving average method
3.Least square method
   

Index numbers,tests of consistency(TRT and FRT) and CPI/CLI(Basics) - uses, limitations, types and formulae

Index number is a measure to record changes in 1 or 2 variables(current and base year quantities) over a period of time.

Uses of Index numbers:
1.To measure changes in the value of money.
2.To measure the cost of living.
3.Helps to formulate economic policies.

Limitations of Index numbers:
1.Index numbers are mere indications of relative changes.
2.The purpose of using a formula can be different in many situations, which leads to a confusion.
3.One of the major drawbacks is that each average is specialised and can be used only in certain cases considering the limitations of the average.

Types of Index numbers:
1.Price index number:It measures the relative changes in the price of a commodity between two periods.
2.Quantity index number:It measures the relative changes in the quantity of a commodity consumed between two periods.
3.Value index number:Measures the change in a nominal value relative to its value in the base year.

Methods of calculating the index number:

1.Simple
 -Simple Aggregate method : A
 -Simple Average of price relatives : B
2.Weighted
 -Weighted Aggregate method : C
 -Weighted average pf price relatives : D

Before we get into the formulae, let's know a few abbreviations:
P01; here P is the price relative, 0 is the base year and 1 is the current year.
N is the number of observations.

Just to add on to the detail;
 v01 = Σp1q1/Σp0q0

Tests of consistency;
1.TRT-Time reversal test
P01 * P10 = 1
TRT is not satisfied by Laspeyre's price index and Paache's price index, but it's satisfied by Fisher's price index.
2.FRT-Factor reversal test
P01 * Q01 = V01
FRT is satisfied only by Fisher's price index.

We can notice that Fisher's price index satisfies both time reversal and factor reversal test. This is one of the reason why Fisher's price index is known as the ideal index number. The other reason is that this index considers both the current and base year quantities.

Consumer Price Index is also known as the cost of living index.
It represents the average change in price over a period of time, paid by a consumer for a fixed basket of goods and services.

Uses of CPI:
1.It indicates the changes in the consumer prices.
2.It evaluates the purchasing power of money.
3.It is also used for comparison purposes.

When we talk about the limitations of CPI;
1. CPI focuses on a fixed basket, as consumer behaviour cannot be predicted, we can't be very sure about CPI value to be relevant.
2. Quality is not considered while calculating the CPI.
3. Inflation effects are not taken into consideration as the basket is fixed.

 

CPI can be computed using 2 methods:
1.Aggregate Expenditure method                                                                                              
CPI = (Total expenditure in current year/Total expenditure in base year)*100;
which means;
CPI = Σp1q0/Σp0q0 * 100
2.Family Budget method
CPI =  ΣWP/ ΣW
where P = p1/p0 * 100

correlation and regression(basics) - types, uses/importance and formula

Correlation is the degree and type of relationship between 2 or more quantities or variables in which they vary together over a period of time. Correlation is represented by 'r'.

Importance or uses:
1.The degree of variation between the variables is to be known for an effective evaluation.
2.It is very much used in forecasts.
3.Correlation analysis is used to understand the economic behaviour.

Types :
Positive and negative correlation.
Linear and non-linear correlation.
Simple and partial correlation.

Positive correlation is when the value of 2 variables change in the same direction.
Negative correlation is when the value of 2 variables change in the opposite direction.
Linear correlation is when there is a constant change in one variable due to a unit change in other variable over the entire range of values.
Non-linear correlation is when there is no constant change due to unit change in other variable over the entire range of values.
Simple correlation is when only 2 variables vary.
Partial correlation or multiple correlation is when more than 2 variables vary.

Main methods of correlation analysis:
1. Scatter diagram
2. Karl Pearson's coefficient of correlation
3. Spearman's rank correlation coefficient

Scatter diagram is the simplest method to calculate the correlation. Under this method, the values for each pair of a variable is plotted on a graph in the form of dots.
Looking at the scatter of several points, the degree of correlation is ascertained, the following are the different degrees correlation:
1.Perfect positive correlation; it is when the value of r = +1.
2.Perfect negative correlation; it is when the value of r = -1.
3.High degree positive correlation; it is when the value of r is positively high.
4.High degree of negative correlation; it is when the value of r is negatively high.
5.No correlation; when the value of r is equal to 0. This happens when the points are haphazardly scattered over the graph and do not show a specific pattern.

Karl Pearson, a British statistician, has given two main formulae to calculate the correlation.
One as a direct method and another which is known as the assumed mean method.

where; 
dx = deviation from assumed mean = x - A

The third method of computing correlation(Spearman's rank correlation) is developed by a British psychologist named Charles Edward Spearman. Here, correlation is calculated under 2 situations:
1.When the ranks are given
2.When equal ranks are given

Regression analysis is a statistical technique used to examine the relationship between one dependent and one independent variable. It can be used to predict the dependent variable when the independent variable is known.

When we talk about the uses of regression analysis;
1.It mainly focuses on business activities and forecasts.
2.To find any unknown variable.
3.The analysis is used to have a check on the quality control.

Before we go into detail, we must know the differences between correlation and regression.
Correlation is used to determine the degree of relationship; whereas regression is used to study the cause and effect of relationship.
When it comes to the correlation analysis, there is no need of independent and dependent variables; whereas in the case of regression analysis, identification of independent and dependent variables is a must.

There are 2 sets of regression equations:
1.x on y; this equation is used to find out the values of x for the given values of y.
    x = a + by
   Σx = na + bΣy
   Σxy = aΣy + bΣy²  
2.y on x; this equation is used to find out the values of y for the given values of x.
   y = a + bx
   Σy = na + bΣx
   Σxy = aΣx + bΣx²   

Where dx = x - A; dy = y - A

Regression coefficients
2 cases;
1.x on y
(x - x̄) = bxy(y - ȳ)
2.y on x                                      
(y - ȳ) = byx(x - x̄)                                     


Regression coefficient using correlation
 2 cases again;
1.x on y
2.y on x

Measures of dispersion(BASICS)

Measures of dispersion are those used to show the degree of variation of the data from its central value.

These measures are of importance because it is used to test the reliability of an average, to be certain of any existence of a high variation in the data, to estimate trends and also for comparison purposes.

There are 4 measures of dispersion :
1. Range
2. Mean Deviation
3. Quartile Deviation
4. Standard Deviation



Range is the difference between the largest value and the smallest value in the data.
It is denoted by R.

R = L-S
where L is the largest variable and S is the smallest.
coefficient of R = (L-S) / (L+S)

Mean Deviation is the value computed by taking the arithmetic mean of the absolute values of the deviations of the functional values from some central value of the data(the mean or the median or the mode).It is denoted by MD.

where; N is the total frequency, f is the frequency and m is the mid value.
coefficient of 
1.MD from mean = MD / x̅ 
2.MD from median = MD / M
3.MD from mode = MD / Z

Quartile Deviation is half of the difference between the third and the first quartile.
It is also known as the semi quartile range. It is denoted by QD.

QD = [(Q3) - (Q1)] / 2
where Q3 is the third quartile and Q1 is the first.
coefficient of QD = [(Q3) - (Q1)] / [(Q3) + (Q1)]  

Standard Deviation is again a measure of dispersion.It is the value computed by obtaining the square root of variance by determining the variation between each data point relative to the mean. It is denoted by SD or σ.

coefficient of σ = σ / x̅
coefficient of variance = σ / x̅ * 100.

The standard deviation is the square root of the variance.

Trigonometry(BASICS)

Trigonometry is nothing but the study of triangles and it's angles .

A more precise meaning :Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.

Before getting into detail let us first understand it's uses in a practical situation.


1. The calculus is based on trigonometry and algebra.
2. The fundamental trigonometric functions like sine and cosine are used to describe the sound and light waves.
3. Trigonometry is used in oceanography to calculate heights of waves and tides in oceans.
4. It used in creation of maps.
5. It is used in satellite systems.

Notation of angles 

The following Greek letters are used generally in trigonometry to indicate in a general way the number of degrees in angles;

theta (θ), phi (ϕ), alpha (α), beta (β), gamma (γ), etc.


Labeling the sides of a right-angled triangle


Hypotenuse is the the longest side of a right-angled triangle, opposite the right angle.
Perpendicular is the side opposite to the angle ( here in the diagram it is the θ ). It is also referred the opposite.
Adjacent side is the side next to the angle in question, the one other than the hypotenuse.
Generally it is the base of the triangle. 



Trigonometrical ratios (t-ratios)

In a right angled triangle ,the three sides which gives six ratios, which are known as trigonometrical-ratios.
1.The ratio of the perpendicular to the hypotenuse is called sine of angle θ and it is written as sin θ.
sin θ = perpendicular / hypotenuse 
2.The ratio of the base to the hypotenuse is called the cosine of angle θ and it is written as cos θ.
cos θ = base / hypotenuse
3.The ratio of the perpendicular to the base is called the tangent of angle θ and is written as tan θ.
tan θ = perpendicular / base

Reciprocal ratios

The following three are the reciprocals of the above ratios:
4. cosec θ = Hyp / Perp
5. sec θ = Hyp / Perp 
6. cot θ = base / Perp

The above ratios shows that;
 sinθ.cosecθ = 1
 cosθ.secθ = 1
 cotθ.tanθ = 1