Elementary row operations

A matrix is an arrangement of expressions, defined in general terms. The items that are arranged are called as elements. We repeat that it is only an arrangement; thereby a matrix does not suggest any algebraic operation between the elements. Due to this fact a matrix can undergo certain operations with its rows, called as matrix elementary row operations or simply as row operations. The row operations are also called as row transformations. Such transformations can be done on columns also.  Let us study the elementary row operations one by one.
1) In a matrix, a row or a column can be interchanged. For example,

a   b   c     can be interchanged as, d e f  or,  as,  a c b                                                                                                              d   e   f                                   

a   b   c             

d   f   e                                                                                                                                                                                 g   h   i

i    g   h            

g   i    h 

In the above example, the row interchange isdenoted as R2 <->R1 and the column interchange is denoted as C2 <-> C3.

2) A row or column can be modified by multiplying by a non- zero real number.For example,

a   b   c     can be modified as,   k[a]  k[b]   k[c]    or,  as,  k[a b c]                                                                                                             d   e   f                               d       e     f            k[d  e   f]                                                                                                                                                                                g   h   i                               g        h     i            k[g  h  I ]

where, ‘k’ is a non-zero real number. These transformations are respectively denoted as R1 -> kR1 and as C1 -> kC1

3) A row or column can be modified by multiplying by a non- zero real number. For example,

a   b   c     can be modified as,   a + kd    b + ke   c + kf      or,  as,   a + kb   b   c                                                                                                              d   e   f                                            d            e          f                        d + ke    e   f                                                                                                                                                                                 g   h   i                                            g             h          i                        g + kh    h   i

where, ‘k’ is a non-zero real number. These transformations are respectively denoted as R1 -> R1 + kR2 and as C1 -> C1 + kC2

These elementary transformations are extremely useful in further topics of matrices, like finding inverse of a matrix.For example, if A is an invertible matrix and B is its inverse, the formulas are,

A = I A and I = BA where I is the identity matrix of the same order.

Start with the equation A = IA.

Plug in the given matrix for A only on the left side and write only the identity matrix times A on the right. That is let the symbol A or the right remain as symbol A.

By repeated elementary row transformations, try to reach in the equation form I = BA. Then the matrix represented by B is the inverse of A.

Understanding of end behavior of functions

End behavior definition:
With respect to polynomial functions, the end behavior of the function refers to the direction and position of the two ends of the function. For example the graph of the function y = x^2, would be as follows:

Note that both the ends of the graph are going upwards. However if we see the graph of the function y = x^3, it would look as follows:

In the above graph, both the ends of the graph point in different directions.

End behavior of polynomial functions:
Polynomial functions have primarily 4 types of end-behaviors. They are as follows:
(a) Both ends going up.

(b) Both ends going down.

(c) Left end going up and right end going down.

(d) Right end going up and left end going down.

It is possible to know the end behavior of a polynomial function without actually graphing it. We use the end behavior model for that. The following are the end behavior rules that help us model the end behavior of polynomial function.

(1) If the degree of the polynomial function is even, then both the ends would point the same direction. Therefore for an even degree polynomial function, both ends would either go up or both the ends would go down.

(2) If the degree of the polynomial function is odd, then both the ends would point in opposite direction. Therefore for an odd degree polynomial function there are two possibilities, either left ends goes up and right end goes down or left end goes down and right end goes up.

(3) If the coefficient of the leading term is positive, then the right end of the polynomial function would always point up. Therefore if the polynomial is of even degree with positive leading coefficient, then both ends would point up, and if the polynomial is odd with positive leading coefficient, then as right end points upward, the left end would point down wards.

(4) If the coefficient of the leading term is negative, then the right end of the polynomial function would always go down. Therefore if it is an even degree polynomial with negative leading coefficient, then both the ends would go down whereas, if it is an odd degree polynomial with negative leading coefficient, then as right end goes down, left end would go up.

End behavior asymptotes:
Polynomial functions do not have asymptotes. However, if it is a rational function, then it would have asymptotes.

Ratios

Ratio is one of the most important concepts in mathematics. Ratio is used to make comparison between two values in mathematics. Ratio in math is represented by colon sign (:). Ratio in words is represented with “is to”. For example: 6 pillows for kids : 3 kids. This expressional can be written or said in word as 6 pillows for kids are to 3 kids.  Ratio can be written in several ways. Let’s have a look at the same in this post along with relevant examples. .

Ratio as a fraction: Ratio can be used as a fraction. For example:
7 apples / 2 kids
13 baby seating chair / 4 kids
30 bed / 2 rooms
Ratio using the word to: Ratio as mentioned above cam be used with the word “is to” or “to”.

For example:
8 baby seating chair is to 2 kids. Therefore each kid will get 4 baby seating chair .
Mohan will buy 3 baby beds online for his 3 nephews. Here, 3 baby beds online is to 3 nephews.
Shreya will buy 5 baby beds online and 2 baby beds from market. Therefore, 5 are to 2.
Ratio using colon: Ratio using colon is the most commonly used form of ratio. It is used to separate two or more values. For example:
8 apples: 4 kids
32: 16: 8: 4: 2
16 boys: 8 girls in the class.
If we multiply each value in a ratio with a non-zero number will result an equal ratio. For example: 2: 4 = 1: 2; 16: 8 = 2: 1; 45: 9 = 5: 1 and so on. Here, 2: 4 and 1: 2 can be referred as equal ratios.
These are some basics about ratio and its usage in mathematics.

Fraction and its types

Fraction is one of the most important concepts in mathematics. When an object is divided into two parts, then each part is called a fraction. For example: Two thirds of the class wanted phone for kidsas gifts. This can be written as 2/3 of the class wanted phone for kids as gifts. Let’s have a closer look at fraction and its types in this post.

Numerator and Denominator :

Every fraction has a numerator and a denominator. Numerator represents the number of parts in the fraction and denominator represents the number of parts in the whole object. For example: 5/6 of toys for infants are made considering hygiene and safety. Here, 5 is the numerator that represents the part of fraction and 6 is the denominator that represents the whole object, that is the number of Baby toys.

Fractions can be classified into three types – proper fraction, improper fraction and mixed fraction.
Proper fraction: When the numerator is smaller than the denominator, it is called a proper fraction. For example: 4/5 has spoken in favor of rechargeable flashlight . Here, 4 less than 5 and therefore, it is a proper fraction.

Improper fraction: When the denominator is smaller than the numerator, it is called an improper fraction. For example: 8/7 has spoken in favor of rechargeable flashlight . Here, 8>7 and therefore, it is an improper fraction.

Mixed fraction: Mixed fractions are formed with a whole number and a fraction. For example:1 2/4. Here, 2 is the numerator, 4 is the denominator and 1is the whole number.
these are some basics on fraction. Let’s do some exercises.

Find the type of fraction from the following
• 2/3:  Proper fraction
• 4/5:  Proper fraction
• 11/2: Improper fraction
• 17/6:  Improper fraction
• 2 4.5: Mixed fraction
These are the common facts about fraction in mathematics.

All about circles

General equation of a circle:
The standard equation of a circle is an equation that when plotted on the co-ordinate plane results in a circular curve. The equation is as follows:
(x-h)^2 + (y-k)^2 = r^2
Where, (h,k) is the centre of the circle and r is the radius of the circle. For example if the centre of the circle is (2,-3) and radius is 4. Then the equation of the circle would be:
(x-2)^2 + (y+3)^2 = 16
The graph of the above circle would look like this:

Standard equation of circle:
A circle with centre a origin and radius r is called a standard circle. Therefore the equation of such a circle would be:
X^2 + y^2 = r^2
Where r = radius of the circle.
For example the standard equation of a circle with radius 3 would be:
X^2 +y^2 = 9
The graph of such a circle would look as follows:

Parametric equation of a circle:
If the parameter is an angle of measure t, then the parametric equation of a circle is given by:
X = r cos(t) and y = r Sin(t), where r = radius of the circle and the angle t ranges from 0 to 2 pi. For example the parametric equations of a circle with radius = 2 would be:
X = 2 cos(t) and y = 2 sin(t).
To be able to find the equation of a circle we need to know two things primarily, the centre of the circle and the radius. However for the standard equation only the radius need be known.

Circumference of Circle Equation:
Suppose there is a circular path around a lake. If we start moving from one point on the path in one direction, then since the path is circular we shall come back to our starting point. The distance thus covered by us in making this trip is called the circumference of the circle. There is no accurate measure of centre of a circle. If we wish to measure the circumference of the base of a cylinder, we can wind a thread around it, and then measure the length of the wound portion of the thread. However this practical method is not too useful for theoretical calculations. Moreover if we wish to find the circumference of very large cylindrical silos then this method is not even practical. The empirical formula derived by mathematicians for circumference of a circle is as follows:
C = 2pir, where r = radius of the circle.

Relations and functions

In Math Relations and Functions play an important role. A set of an ordered pair is termed as the relation. So a relation just signifies the relation between the x and y value in a function.

We can say that relation is just the set of ordered pairs. A function on the other hand is the set of ordered pair in which for every value of x, there is one value of y. How to do Relations and Functions–To do the relations and functions, we express them in brackets just like {x, y}.

The first element of the bracket is x and the x values forms the domain of any function or a relation. The second element of the brackets is y and the y values forms the range of any relation or the function. Math Functions and Relations go hand in hand.

The relations are not functions but a function shows the relation between x and y values. Examples of Relations and Functions are given below: - (1,3) , (2,3) , (5,6) , (7,9). This is a relation consists of ordered pairs where domain is the set {1, 2, 5, 7} and range is {3, 6, 9}.

This signifies a relation and not a function as for both the values 1 and 2, the range is 3. But in a function, there should be just one value of y for every x value.
Graphing Relations and Functions – We can graph the functions and relations, by simply plotting the x values on the x axis and the y values on the y axis. The only difference between a relation and a function is that x value in a function has one and only one y value.

To check whether the given set is a function or not, we can do the vertical line test, in which we check that vertical line drawn through the graph will intersect the function at one point only. Hence if there is a vertical line for each pair then that means it is a function and if not that means it is a relation.

So we can conclude that a function looks like a relation only but the only difference is that it is loyal to its partner that means for every x value in the pair there will be just one y value.

Step Functions

In our day to day life we always experience step functions but we do not notice it mostly. If you have noticed about the calling rates of cell phone, as it is charged per minute. You generally pay a certain amount of money for the first minute. And as soon as we run over to the second minute you start paying. So let it be say, you have 1 min 2 sec. you pay the same. For example, if we understand by using an example, say a cell phone call rate is charged 40 cents per min. as soon as we go over 1 minute by 1 or 2 seconds we are charged double for the another minute as well that is another 40 cents been charged likewise we have to pay in all together 80 cents up to 2 minutes of a phone call. This was the brief example to understand the step function definition. When we graph it, it looks like steps. We notice on the graph, first minute of 0 to 1, 40 cents. And then as soon we hit 2, one second after 1 minute. We are charged 80 cents.

The step function graph looks like a staircase. What is the integer graph and how does it affects the function graph. F(x)=[(x)], it is a greatest integer function. It says nothing but give the greatest integer or less than or equal number in it. Let us say for example, [(3.7)] so as we know integers are all sort of whole numbers negative or positive. Here we are looking for the greatest or less than the or equal to 3.7. So 3.7 is not an integer by its own. So we have to go to the next smallest one, which is down to 3. So, [(3.7)] =3. Now what if the [(8)], here 8 is already an integer, so the greatest integer, less than or equal to 8 is 8. So [(8)]= 8.  Now what if the number is negative, say [(- 2.5)] here if we think about how we do it on a graph using step function on a number line. Step functions algebra 2 , we had studied that say like on a number line 3.7 is on the right side, we go down to 3. Then 10.2 on the number line , but we are going down to 10.but on number line – 2.5,we will not change the direction towards right side as we did in positive integers. But we end up going on -3.as we see on number line -3 is smaller than the -2.5.