Dot Product of Two Matrices

Introduction about dot product of two matrices:

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually denoted as the coordinate vectors) and returns a single number obtained by multiplying corresponding entries and adding up those products. The name is derived from the dot "·" that is often used to designate this operation. Here the dot product of the two matrices are represent the product of two matrices that give finally one matrix as result. Understanding Derivative Product Rule is always challenging for me but thanks to all math help websites to help me out.

(Source from Wikipedia)

Explanation about "dot Product of Two Matrices "

Two matrices S having order x x y and D having order a x b are said to be conformable for the process ofmultiplication if the columns count of the first matrix S (y)  is equal to the rows count of the second matrix B(a).

Then the order of SD is x × b =rows count for matrix A × columns count for the matrix B

On considering  the matrices given  having same rows and columns, perform the multiplication operation
`[[-1,-2],[-1,-4]]`  and   `[[-1,1],[-1,1]]`

Solution:

We take the first matrix as G and G= `[[-1,-2],[-1,-4]]` has the order  of  2 x 2.

And take the second matrix as H and H= `[[-1,1],[-1,1]]` has the order  of  2 x2

For multiplication operation on G x H

Here the orders present are  2 x 2  in G and 2 x 2  in H so it is possible

G x H =   `[[-1,-2],[-1,-4]]` `[[-1,1],[-1,1]]`

=  `[[1+2,-1-2],[1+4,-1-4]]`

= `[[3,-3],[5,-5]]`

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Problems to Explain "dot Product of Two Matrices "

On considering  the matrices given  having same rows and columns, perform the multiplication operation
`[[32,-11,-13,0],[-11,10,-11,-11],[-11,-11,10,-11]]` and `.``[[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0]]`

Solution:

We take the first matrix as G and G= `[[32,-11,-13,0],[-11,10,-11,-11],[-11,-11,10,-11]]` has the order  of  3 x 4.

And take the second matrix as H and H= `[[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0],[0,0,0,0,0]]`    has the order  of  5 x5

For multiplication operation on G x H

We must  have equal value on the columns for the 1st matrix 'G' and the rows of the 2nd matrix 'R'.

We have the columns of the 1st matrix 'S' is 4  with  the rows of the 2nd matrix 'R' is 5.

So, the operation on the multiplication is not comfortable.

Types of Events

Introduction to types of events:

The word ‘event’ literally means ‘happening. This is an accepted fact that anything may happen at any time. But, some are preset and some are instant. Some are the markers of fortune and others are the markers of misfortune, According to types of the occurrence, events can be categorized as special, temporal and accidental. Please express your views of this topic Probability of Compound Events by commenting on blog.

Types of Events-special Events
Special events are marked as the renewable variety. This type of events is remembered by all. The examples of such events are birthday, marriage anniversary, the moment of meeting someone who has a great impact on one’s life, or the day of receiving some award. These days have a lasting influence on one’s life and these are the major sources of inspiration in future.

Types of Events-temporal Events

A category of events can be made on the basis of the time of an action. It can be called ‘temporal events’. The temporal events may be of three types: past, present and future. The past action points at the event happened before and may be working on the action that is happening now, as we call it a present action. The relation between the past and the present can be explained as the cause and the effect. For example, ‘one studied well and he passed a test very well.’ Here, ‘the activity of study’ is the past event and ‘passing the test’ is the present event. And, based on the present events the career options or future study options are chosen and that choice is a future event as it is to happen next to the present event. So, in a sense, the past, the present and the future events are related by cause and effect. I have recently faced lot of problem while learning live online math help, But thank to online resources of math which helped me to learn myself easily on net.

Types of Events-accidental Events

There are a few events that are not preset or programmed but fully based on chance. The events happening by chance has a positive or negative outcome. The events may be like, winning a lottery makes a happy end and it may be a mishap marking a tragedy or a sad episode. This two-pronged intersection is an important junction in the eventful lifestyle whose basis is experience or accident which is either special or disappointing.

Statistics Poisson Distribution

Introduction to statistics Poisson distribution

The Poisson distribution in statistics is named for the French mathematician S. D. Poisson. It is used to describe a number of processes like the distribution of telephone calls going through a switchboard system, the demand of customers for service at a restaurant, the arrival of customer at a shop and the number of accidents at a junction.

Characteristics of a Statistics Poisson Distribution

1. The experiment consists of counting x the number of  times a particular event occurs during a given unit of time.

2. The probability of the occurrence of  an event in a given unit of time is the same for all units.

3. The number of events that occur in one unit of time  is independent of the number that occur in other units.

4. The mean number of events in each unit will be denoted by the Greek letter `lambda` .

Statistics of Poisson Distribution-mean and Variance

The probability distribution :

P ( x )  = ( `lambda` x e -`lambda` ) / x !

where `lambda` = mean umber of events during the given time period,

e = 2.71828 ( the base of natural logarithm )

the mean `mu` = `lambda`

the variance `sigma` 2 = `lambda`

Having problem with Determining Sample Size for Research Activities keep reading my upcoming posts, i will try to help you.

Example of Statistics Poisson Distribution

Let us investigate the safety of a dangerous curve. Past police records indicate a mean of 5 accidents per month at this curve. Suppose the number of accidents is distributed according to a Poisson distribution. Calculate the probability in any month of exactly 0, 1, 2, 3 or 4 accidents.

Solution : Since the number of accidents is distributed according to a Poisson distribution and the mean number of accidents per month is 5, we have the probability of accidents happening in any  month

p ( x )  = (` 5 ^ x e ^ ( -5 )) / x !`

`By this formula, we can calculate`

`p ( 0 ) = 0.00674 p ( 7 ) = 0.104445`

`p ( 1 ) = 0.3370 p( 8 ) = 0.065278`

`p ( 2) = 0.08425 p ( 9 ) = 0.036266`

`p ( 3 ) = 0.14042 p ( 10 ) = 0.018133`

`p( 4 ) = 0.17552 p ( 11 ) = 0.008242`

`p ( 5) = 0.175467 p ( 12 ) = 0.003434`

p ( 6 ) = 0.146223

Solve and Check Inequalities

Introduction to solve and check inequalities:

Inequality is a one the concept of algebra.Inequality in the form of equation with greathearted or lesser than symbol. . Inequalities are described with the symbol of <, >, <=, and >=It is algebra expression. Inequalities is a combination of variables, numbers ,constants , conditions( < or >) with operations( + or -)

Four forms of checking inequalities:

1. Ax + By  <  C

2. Ax + By  >  C

3. Ax + By  <  C

4. Ax + By  >  C

Basic Concepts of Checking Inequalities:

Types of inequalities with examples

Less Than (<)

Example of solve and check inequalities:

x<6 br="br">
Here variable x is less than the value of six ,It mean x may contains the value of like 5,4,3,2,1,0,-1,-2,-3 … etc

Less Than (≤)

Example of solve and check inequalities:

x ≤ 8,

Here  variable x is less than or equal to 8 mean , that is x may contain the value of 8,7,6,5,4,3,2,……….. so on.

Greater than (>)

Example of solve and check inequalities:

x > 10,

Here the variable x  is great than value of  10, that is x may be any one of the following value 11,12,13……so on.

Greater Than or equal to (≥)

Example of solve and check inequalities:

x ≥ 3,

Here  variable x  is great than or equal to 3 means that, that is x may be contain any one of the following value 3,4,5,6……so on.

Example Problems in Inequalities:

Solving and check the inequalities

1. solve and check: 4x+5 < -2x+14

Solution:

4x+5 < -2x+14(given problem)

4x+5-5 < -2x+14-5( subtract both sid e by 5)

4x < -2x+9

4x+2x < -2x+2x+9( add both side by 2x)

6x < 9

6x /9< 9/9( divide both side by 9)

x < 3/2  in otherwords x<1 .5=".5" br="br">
So x values are may 1.4.1.3,1.2,1.1,1…..and so on

Check the ineuqlities:

Substitute x value to the given problem

Substitute x=1 in  4x+5 < -2x+14

4(1)+5 < -2(1)+14

5< -12

Condition are not satisfied here 5 is graeterthan the -12

So given problem is not inequalities.

Exampl 2:

Solve  and check x + 3 < 2x +7.

Sol:

x + 3 < 2x + 7

x +3 – 3 < 2x + 7 – 3 (Rule 1)

x < 2x + 4

x-2x < 2x-2x+4

- x<4 br="br">
x > – 1

The solution set is {0, 1, 2, 3…}

Verification:

Substitute x =0,1,2

Substitute x = 0 in given eqaution

X+3<2x br="br">
0+3<2 br="br">
3<7 br="br">
Here condition is satisfied so given equation is inequality

Discrete Math Counting

Introduction for discrete math counting:

Counting means the process of a counting, the number of samples or number of outcomes or number of probable ways to do a thing. The counting techniques are permutation, combination and factorial. Permutation means the process of rearranging the given number of objects or elements. Combination means the process of selection of elements or objects from a collection. In combination order of elements is irrelevant. Factorial means the value of a number is the product of a given number and all smaller positive numbers. Let us prepare for a discrete math counting techniques formulas and example problems.I like to share this Permutations and Combinations Word Problems with you all through my article.

Formulas for Discrete Math Counting:

1) Formula for Permutation:

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

2) Formula for Combination:

C(n,r) = `(n!) / (r!(n-r)!)`

3) Formula for Factorial:

n! = `n*(n-1)*(n-2)...3*2 *1`. Is this topic binomial probability distribution hard for you? Watch out for my coming posts.

Example Problems for Discrete Math Counting:

Example problem 1:

How many ways 3 mouses can be taken from among 8 mouses?

Solution:

Permutation: P(n,r) = `(n!) / ((n-r)!)`

Here, n = 8, r = 3

P(10, 6) = `(8!)/((8 - 3)!)`

= `(8xx7xx6xx5!) / (5!)`

After simplify this, we get

= 336

In 336 ways 3 mouses can be taken from among 8 mouses.

Example problem 2:

In how many ways 5-chairs can be arrange?

Solution:

Factorial: n! = `n*(n-1)*(n-2)...3*2 *1`

5! =  5 x 4 x 3 x2 x 1

After simplify this, we get

5! = 120

Example problem 3:

How many ways 2 books can be chosen from among 10 books?

Sol:

Combination: C(n,r) = `(n!) / (r!(n-r)!)`,

Here n = 10, r =2

C(10, 2) = `(10!) / (2!(10 - 2)!)`

= `(10xx9xx8!) / (8!xx2xx1)`

After simplify this, we get

= 45

In 45 ways 2 guides can be chosen from among 10 guides.

Example problem 4:

In how many ways 6 lights can be arrange?

Solution:

By using the Factorial formula

: n! = `n*(n-1)*(n-2)...3*2 *1`

6! = 6 x 5 x 4 x 3 x 2 x 1

After simplify this, we get

6! = 720

The above examples are helpful to study of discrete math counting techniques.

Prime Example Definition

Introduction to prime example definition:

In this section we have prime example definition. Prime numbers are divisible by itself and one. The non prime numbers are known as composite numbers. We are going to solve some based on the prime numbers. Let us study about prime example definition with some solved problems along with step by step answer and exercise problems.

Example Problems for Prime Example Definition:

Example problem 1: What is the prime factorization of 10?

Solution:

Given number is 10

Divide by prime factors until the quotient is 1.

10 ÷ 2 = 5

5 ÷ 5 = 1

The prime factorization of 10 is: 10 = 2 × 5

Answer: The prime factorization of 10 is: 10 = 2 × 5

Example problem 2: Is 37 a prime number or composite number?

Solution:

Given number is 37

Prime numbers are divisible by itself and one.

37 is divisible by 37 and 1 only.

Therefore, 37 is prime number.

Answer: 37 is prime number.Please express your views of this topic Roman Number by commenting on blog.

Practice Problems for Prime Example Definition:

Practice problem 1: Can you tell the prime factorization of 2?

Practice problem 2: Write the prime factorization of 54. Use exponents when appropriate and order the factors from least to greatest (for example, 22 × 3 × 5).

Practice problem 3: Which of the following numbers are not prime numbers? 12, 37, 88, 90, and 101

Practice problem 4: Is 727 a prime number or composite number?

Practice problem 5: Write the prime factorization of 6. Use exponents when appropriate and order the factors from least to greatest.

Solutions for prime example definition:

Solution 1: The prime factorization of 2 is: 2

Solution 2: The prime factorization of 54 is: 2 × 3 × 3 × 3

Solution 3: 12, 88, and 90 are composite numbers.

Solution 4: 727 is a prime number.

Solution 5: The prime factorization of 6 is: 2 × 3