Introduction to non invertible matrix

Introduction to non invertible matrix:

Many people talk about matrices but do you know what exactly a matrix is? Let's see. A matrix is a set of numbers arranged in a particular order in a rectangular array with  m rows and n columns in it.Below is an example of matrix in which a_ij are know as elements or entities.

Matrices are of different types. They are

• Row Matrix
• Column Matrix
• Square Matrix
• Rectangular Matrix
• Diagonal Matrix
• Scalar Matrix
• Identity Matrix
• Null Matrix

Define Non Invertible Matrix:

A matrix is said to be non-invertible matrix if and only if its determinant is zero.Non-Invertible matrices are also known as Singular matrices.They are also known as Degenerate Matrices.

Conditions for the matrix to be non-invertible are

1. It should be a square matrix,i.e.the number of rows and columns in a matrix should be equal.
2. The determinant of square matrix should be zero.

Que : How to calculate the determinant of a matrix?

Ans : Step1 - Let  A be a 2 x 2 matrix

Step 2 - Multiply the elements in the first and second diagonals and subtract the product of the second diagonal from the first.

Formula for finding the value of the determinant = `a_(11)` *`a_(22)`  -  `a_(12)` *`a_(21)`

det A = 4 * 5 - 2 * 10

          = 20 - 20

         = 0

Example Problems Based on Non Invertible Matrix:

Ex : 1 Calculate the determinants for the following matrices and check if they are non-invertible matrices.

 `[[5,6],[2,15]]`

Sol : Step 1 - Let  A = `[[5,6],[2,15]]`

Step 2 - Formula = `a_(11)` *`a_(22)`  -  `a_(12)` *`a_(21)` 

det A = (5x15) -(2x6)

detA = 75 -12 = 63 . Since det A is not equal to zero, the given matrix is not a non invertible matrix.

Ex: 2 : Calculate the determinants for the following matrices and check if they are non-invertible matrices.

 `[ [1,0,0],[-2,0,0],[4,6,1]]`    

Sol :  Step 1 : Let A be the given matrix.

Formula : `a_(11)` *`a_(22)`  -  `a_(12)` *`a_(21)` 

Step 2: Plugging in the values in formula

det A = 1[(0 * 1) - (6 * 0)] - 0[(-2 * 1) - (4 * 0)] + 0[(-2 * 6) - (4 * 0)]

det A = 0 - 0 + 0

det A = 0

Since det A is equal to zero ,the given matrix is a non-invertible matrix. 

Sample Variance

Definition of sample variance:

In 1893 the famous statistician Karl Pearson gave the definition of standard deviation as a measure of dispersion and gave a method for its calculation. Of all measures of dispersion the standard deviation is most commonly used. To understand what is sample variance, let us first see what is sample standard deviation.
Standard deviation: The positive square root of the number, obtained by dividing the sum of the squares of deviations from mean of observations of given data, by the number of observations is called the standard deviation of the data. It is denoted by s. Thus if x ¯ is the mean of n observations x_i (i = 1,2,3,…n), then,

With this background let us now define sample variance:
It is convenient to calculate the standard deviation in two steps. In fact, the number obtained by dividing the sum of the squares of deviations from the mean of observations of the given data by the number of observations is called the variance of the data. The variance is denoted by s^2 or s^2. After calculating the variance, its positive square root gives the standard deviation s. Thus, based on the above sample variance definition, we can write the sample variance formula as follows:

Calculation of variance for ungrouped data:

Direct method: Calculate the mean x ¯ of observations x1, x2, x3, … , x_n. Find the deviation x_i - x ¯ for each x_i. Calculate the square (x_i - x ¯)^2 for each x_i. Find the sum ?(x_i - x ¯)^2. Divide ?(x_i - x ¯)^2 by n to get the variance s^2.

If the values of x_i are relatively small and the mean is a fraction or a lengthy decimal, then we have an alternative formula for variance as follows:

Because of the presence of x_i^2 in the alternate formula for variance, finding variance by this formula could be tedious when x_i^2 are relatively large. The first method is also time consuming if the mean is a fraction (and not an integer). Fortunately, the numbers to be handled can be made small and the fractional mean can be avoided in calculations, by taking deviations of x_i^2  from an appropriately chosen number A, called the assumed mean.

Trigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships.

Trigonometry has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. It is usually taught in secondary schools either as a separate course or as part of a precalculus course. Trigonometry is informally called "trig".

A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation
Let's see one of the trigonometry examples
Question:-

solve for x

2 sin²x- sinx-1=0

Answer:-

We may use some trigonometry formulas to solve this.
Put sinx = z in the given equation ,you get

2z²-z-1 =0

2z²-2z+z-1=0

2z(z-1)+1(z-1)=0

(z-1)(2z+1)=0

either z-1=0 or 2z+1=0

so z=1 or z= -1/2

since z = sinx we get

sinx = 1 = sin π/2 or sinx = -1/2 = sin(π+π/6)

x= π/2 or x= π+π/6

so x= 90° or 210°

Help with triangles and angles

Introduction of understanding triangles:
In this article let me help you on triangles and angles. Triangles is the important geometrical shapes, it is a closed shape with three sides and angles. The triangles are divided into two categories they are based on sides and angles. Comparing sides the triangle consist of three types they are equilateral triangle, scalene triangle, and isosceles triangle.

Comparing angles we consider the triangle as three types they are right angle triangle, acute angle and obtuse angle. In this article, we will understanding triangles with suitable example problem. For more Study material.
Understanding Triangles in Sides
Understanding triangles in sides are classified into three types they are show below.

• Equilateral triangle
• Scalene triangle and
• Isosceles triangle

Keep reading may be in the next session let me help you on multiplying mixed fractions

Help with algebra worksheet

Introduction to algebra worksheets

Halfway Algebra follows the rules of arithmetic in which letters representing book. The amount is the not changing the see. Mediate Algebra follows the rules of arithmetic operations, and relationships conjunctive the drawing.

Mediate Algebra uses mathematical equations to draw relationships between elements that motley over term; These elements are called as variables.In this worksheets, there are more solved problems. These problems are rattling cooperative to bang nigh work intermediate algebra. This could also help us on statics examples

Let us see gray algebra properties and representative problems , worksheets.

Keep reading may be in the next session let me help you on rules of logarithms

Equation solver online

Introduction to derive equation of motion:
In this article let me help you on equation solver online. Equation solver describe the behavior of a system or the motion of an object or a particle under the influence of a force as a function of time. These famous equations of motion in elementary Classical

Mechanics are given below:
v= u + at
s =( u + v ) t /2
s = ut + at2/2
v2 = u2 + 2as

The following are the variables used in these equations:

* t is the length of the interval of time and measured in second.
* u is the initial speed of the objects or the speed at the beginning of the time interval, measured in ms-1.
* v is the final speed of the objects or the speed at the end of the time interval, measured in ms-1. This could also help us on homogeneous mixture. This could also help us on linear definition
* a is the accelerations of the object within the time interval t measured in ms-2.
* s is the distance covered or the displacements of the object during the time interval t, measured in metres.

Keep reading may be in the next session let me help you on fractional exponents.

Algebra 2 book online

Introduction to Saxon algebra 2 help online:

In this lesson let me help you on algebra 2 book online. In algebra 2, there are many symbols are used and the variables are used for the unknown values. In Saxon algebra2 the symbol is used for the variables or the alphabets to denote the number values. In online, the tutor will explain about the Saxon algebra 2 and the students may learn the concepts from the other websites also. Now we see the Saxon algebra 2 through online with the help of tutor.
About the Saxon's Algebra 2 Help Online: This could also help us on parallel line

Let us see about the Saxon's algebra 2 through online with the help of the example problems by the tutor.

Keep reading may be in the next session let me help you you Introduction to trigonometric integrals


I hope you like my information on this article. Keep reading may be in the next session let me help you on