Matrix: Cofactors

Consider a matrix that is a square matrx. That means that the matrx is n X n matrx. The number of rows and the number of columns in such a matrx are equal. Each term of the matrx can then have a cofactor. The position of each of the element of a matrx is described by (i,j). Where ‘i’ is row number (ith row) and j is the column number (jth column). For example consider the matrx below:
[a b c]
[d e f]
[g h l]

Here the position of the element a is (1,1) since it is the first element in the first row and first in the first column. Similarly the position of the element f would be (2,3) since it is the third element in the second row. So row number of f is 2 and column number is 3.

Cofactor of matrix:
To be able to find the cofactors of a matrix, we first need to find the minors of each of the elements. This can be done as follows:
If we take any entry, say ‘a’ in the above matrx and remove the row and the column containing a and keep the other entries in the same order, we get the determinant
|e f|
|h l|

This determinant is called a minor of ‘a’. Thus, removing the column and the row containing a given element of a matrx and keeping the surviving entries as they are, yields a determinant called the minor of the given element.

If we now multiply the minor of the entry in the ith row and jth column by (-1)^(i+j), we get the co-factor of that element. Therefore in the above matrx, the minor of h is
|a c|
|d f|
And multiplying this by (-1)^(3+2) = (-1)^5 = -1, we get the cofactor of the element h. This cofactor, therefore is
(-1) * |a c| = -(af – cd)
           |d f|

Note: The co-factor of an element is precisely the factor by which that element is multiplied in the expansion of the determinant of that matrx.

The co factors of a,b,c,… etc are denoted by Ac, Bc, Cc,… and so on.

If every element in a matrx is replaced by its cofactor, the resulting matrx is called the cofactors matrix. Therefore for our matrx above, the cofactors matrx would look like this:
[Ac Bc Cc]
[Dc Ec Fc]
[Gc Hc Lc]

Where, Ac = +(el-fh), Bc = -(dl-fg), Cc = (dh-eg), Dc = -(bl-ch) and so on.

Solving by elimination method

The elimination method is one of prominent methods in solution to the system of linear equations with two variables. The elimination method for solving equations may be apparent in certain cases where, the coefficients of a particular variable or variables are same.

In such a case you automatically use this technique. These cases of equations are called solving elimination equations. For, example, if x + y = 5 and x – y = 1, you eliminate ‘y’ by just adding both the equations to get the solution as x = 3. Similarly you eliminate ‘x’ by subtracting the second equation from first and figure out y = 2. But all systems are not as simple as this. Hence let us take a broader look on solving by elimination method.

Suppose, you have a system of equations in which all the coefficients of the variables are different. Select a variable for elimination. Your skill and experience will tell you which will be the ideal variable to eliminate. Take the LCM of the coefficients of that variable and multiply each equation with the ‘missing’ factor of the LCM in each case.

Now the given set of equations is transformed in such a way that the new set has the same coefficient for the selected variable in each equation. Now do the subtraction or addition operations on the transformed equations, so that the selected variable is eliminated and you get the solution of the remaining variable. Subsequently, plug in that in any of the given equations to figure out the solution of the other variable.

Let us elaborate with an example. Let the system be 17x + 2y = 49 and 19x + 3y = 54. It is prudent to decide to eliminate ‘y’. Eliminating ‘x’ not an incorrect step. But that will lead to cumbersome working. The LCM of 2 and 3 is 6.

Multiplying the first by 3 and the second by 2 (the missing factors of 6), the equations are transformed to 51x + 6y = 147 and 38x + 6y = 108.. A subtraction operation gives you the result as 13x = 39, means x = 3. Now plugging in x = 3 back, you can figure out y = -1.

The method of solving equations by elimination is the basis for the concept of use of determinants for solution of system of equations.

let us make a simple illustration. Suppose a1x + b1y = c1 and a2x + b2y = c2 . By elimination method you can establish, x = (c1b2 – c2b1 )/(a1b2 – a2b1 ), which is the concept behind Cramer’s rule of solution by using determinants.

Sequence and series

Sequence Series is a very important part of mathematics. When we write numbers in an order then we call it sequence of numbers.

The list of numbers follows a particular trend in that. The numbers in a seq are called terms of the seq. We cannot flip the terms in a seq.

Their order cannot be changed. We do the counting of terms from left to right.
For example: - 1, 3, 5, 7, 9 is a seq of five terms where first term is 1 and third term is 5 and last term is 7.

If we look at this seq we see that this seq follow a certain trend and that is every next term is two more than the previous term. Now what is a series?

A series is when we add the terms of a seq that constitutes to form a series. There are different types of seq and ser that exists and they are: -

1. Arithmetic Sequences and Series– In this type of seq and ser, the next term can be determined by adding the common difference to the previous term. The general arithmetic seq and ser is given as a, a+d, a+2d, a+3d, a+4d… where ‘a’ is the first term and d is the common difference. For example: - 1, 4, 7, 10, 13, 16. Here we see that first term is 1 and the common difference is 3

2. Geometric Sequences and Series– In this type of seq and ser, the next term can be determined by multiplying the common multiple by the previous term. The general geometric seq and ser is given as a, ar^2, ar^3, ar^4… where ‘a’ is the first term and r is the common multiple. For example: - 2, 6, 18, 54. Here we see that the first term is 2 and the common multiple is 3 which can be determined by dividing any term by its previous term.

3. Infinite Sequences and Series– In this type of seq and ser, the last term is unknown to us. We do not have fixed number of terms in that as it is an infinite ser. For example: - 2, 6, 10, 14…..

Series and Sequences Formulas are used to determine the first term, last term, sum of terms, number of terms or the common difference or ratio.

Equivalent Decimals

Decimals are nothing but a whole numbers but broken into pieces, so example we have one whole block, and we cut it in 2 pieces, and we call it as Tenths. How to do equivalent decimals, so how do we write it ? We write it as 1 .2 tenth (Here decimal represents AND). But if we break that down even further into more ten pieces and take out and add 3 then we call these as Hundredths.  So this way we have once, tenths and hundredths. So this way we will write it as 1.13 (one and thirteenth, remember here the last piece that we saw. Here we will name the last 1 and 3 as thirteenth, the last digit even if it is zero we call it that). Now late us take one more piece of cube and break it down into ten equal parts that will be called as thousands. So we have ones, tens, hundredths and thousands. Let us say we have this number up in broken to two pieces .we will call it as 1.232 that is one and two thirty two thousands, so we read the decimal like this way.

Now moving up to equivalent decimals examples, what if we have only one block so we will call it as 0.1( here we will read it as zero and one tenth.) but let us say we want to turn it to hundredths. The whole one piece when broken into ten equal parts has the same value for 0.1 as well. However they are hundredth. so all we need to say hundredth we need to add a zero. Because when we read decimals we need to say the last digit. Thus it will be 0.10(we call it as zero and one tenth one hundredth.) so one tenth is equal to tenth hundredths.

How to find equivalent decimals? We can by taking the denominator and dividing it with the numerator of the same fraction. Now for example we have ½ as a fraction so, 2 have to get divided by 1 so we will add a zero and bring up a decimal point. This way we have 10 which will be divisible by 10. As 2 times 5 is equals to 10.  And 10 subtracted by 10 and remainder comes to zero. So we have got the solution as 0.5 is the equivalent decimal.
What is a Equivalent decimal are nothing but the decimals which have the equal proportions.

Direct Speech and Indirect Speech

Direct Speech and Indirect Speech
Direct and Indirect Speech is one of the very important concepts in English learning. Speech can be classified into direct speech and indirect speech. Let’s have a look at the same in this post along with some examples for better understanding.

Direct Speech
Direct speech is a type of speech that exactly says what someone has said. In direct speech, the exact saying is repeated or quoted and the words spoken are written within inverted commas (“-----“). Direct speech is therefore also referred as quoted speech.

Examples of Direct Speech
1. Uncle John said, “I have bought an infant play gym for my son. It is not just fun but is also healthy.”
2. Mary asked, “Have you tried online diaper purchase?”

Here, in the first sentence, the speech of Uncle John speaking about his buying an infant play gym for his son is exactly presented. On the other hand, in the second sentence Mary’ question to on trying online diaper purchase is repeated exactly.

Indirect Speech
Indirect speech is a type of speech that encloses what a person has said. It doesn’t use quotations and is not repeated exactly like direct speech. In indirect speech, the tense usually changes as when we use indirect speech, we generally speak about something in the past.

Examples of Indirect Speech
1. Roy said that he enjoys to shop baby online products as it is easy and fast.
2. Mohan asked to Riya whether her kid has started going to school.
Here, in the first sentence, the Roy’s enjoying to shop baby online products is reported indirectly and not repeated. On the other hand, in the second sentence Mohan’s question about Riya’s kid going to school is similarly reported and not repeated.

Change direct speech to indirect speech:
1. Tony said, “I enjoy shopping for kids online.”
Tony said that he enjoys shopping for kids online.
2. Maria asked, “Do you eat non-vegetarian food?”
Maria asked me if I eat non-vegetarian food.
3. Father asked, “How long have you worked here?”
Father asked me how long I was working there.

Active and Passive Voice

Active and Passive Voice are two of the most essential concepts in English grammar. Taught in early middle school, active and passive voice is used extensively while speaking English language. Active voice and passive voice are two special forms of verbs. Let’s have a look at active voice definition and passive voice definition along with examples.

Active Voice
When the subject of a sentence performs the action expressed by the verb, the verb form is in active voice. In other words, the object receives the action of the verb. Active voice is most commonly used while speaking English.

Examples of Active Voice:
1. Mrs. Shah gets two baby dresses from  Little Kangaroo Kids wear collection. (Here, ‘Mrs. Shah’ is the subject, ‘gets’is the verb and ‘two baby dresses from Little Kangaroo Kids’ wear collection’ is the object.)
2. Maya shops baby clothing from online newborn baby clothes India stores. (Here, ‘Maya’ is the subject, ‘shops’ is the verb and ‘baby clothing from online newborn baby clothes India stores’ is the object.)
3. Ram sings a song. (Here, ‘Ram’ is the subject, ‘sings’ is the verb and ‘a song’ is the object.)

Passive Voice
When a subject in a sentence is acted by some other agent or something unnamed, the verb form is in passive voice. Passive voice is generally used when the action is more important than the subject in a conversation.

Examples of Passive Voice:
1. Infant clothes online India collection is explored by Mina. (Here, the subject Infant clothes online India collection’ is acted by the other agent ‘Mina’.)
2. Disney toys are loved by kids. (Here, ‘Disney toys’ is the subject, ‘is loved’ is the verb and ‘by kids’ is the object.)
3. Fish are eaten by cats. (Here, ‘Fish’ is the subject, ‘are eaten’ is the verb and ‘by cats’ is the object.

Active to Passive Voice:
1. Rama loves Sita
Answer: Sita is loved by Rama
2. Everybody believes in God
Answer: God is believed by everybody
3. Mother shops baby essentials from online stores
Answer: Baby essentials from online stores are shopped by mother
These are the basics about active and passive voice.

Scientific Notation Definition

Scientific Notations are standard way of writing number which are too small or too big.
In this standard notation all numbers are written in the form of:
m * 10n
here exponent n is an integer, m is mantissa which is a real number.
Scientific Notation Rules are as follows:
The value of mantissa holds a real number from 1 to 9 and exponent n contains the place value to get the original number.
If the decimal shifts to left by ‘n’, then the value of exponent increases by ‘n’. if decimal shifts to right the exponent decreases by 1 for each shift.
For addition or subtraction of numbers in scientific notations form their exponents should be same.
For multiplication of two such numbers, mantissa is multiplied and exponents are added.  
For division exponents are subtracted.

Scientific Notation Problems are as follows:
Q.1) Write the following numbers in scientific notations form:
1) 56788
2) 67.345
3) -6890
4) 0.000009888
5) 978.0 * 10-5
Sol.1)
1) 56788 = 5.6788 * 104. As we know that the mantissa can be real number from one to 9 so we shifted decimal to left four times and hence the exponent of 10 increases by 4.
2) 67.345 = (67.345)*100, in this number, value of mantissa is greater than 1 hence we shift the decimal to left and increase exponent by 1. So its scientific notations form is: 6.7345 * 101
3) -6890 = (-6.890)*103 by using rule used in above solution.
4) 0.000009888 = (0.000009888)*100. This number is less than one hence we will shift the decimal to the write and decrease the decimal by 1 for each right shift.
(9.888)* 10(0-6) = 9.888*10-6.
5) 978.0 * 10-5 is in exponential form but not in standard scientific notations as mantissa is greater than 1. So, shift it to let two time and increase exponent by 2 to get 9.78 * 10-5+2 = 9.78*10-3

Q.2) Solve the following:
1) (7.0  x  102)+(9.4 x 106)
2) (9.4 x 104)(3.5 x 10 –5)
3) (3.5  x  10 – 2)/(9.6 x 10 – 4)
Sol.2)
1) As exponents are not same. 9.4x106 = 94000 x 102
    (3.0  x  102)
  + (94000.0  x  102)
      94003.0 x 102 = 9.4003 x 106
2) (9.4 x 104) (3.5 x 10 –5)=(9.4 x 3.5) x 10(4+(-5))  (exponents are added as multiplication operation)
= 32.9 x 10-1
= 3.29 x 100
3) (3.5  x  10 – 2) / (1.5 x 10 – 4)
= (3.5/1.5) x 10 (-2-(-4)) (exponents are subtracted as division operation)
= 2.33 x 102