Singular Matrices

The term determinant was first introduced by Gauss in 1801 while discussing quadratic forms. He used the term because the determinant determines the properties of the quadratic forms. We know that the area of a triangle by means of vertices (x1, y1) (x2, y2) and (x3, y3) is

1/2 [x1(y2-y3) + x2 (y3-y1) + (y1-y2) ]

Similarity the condition for a second degree equation in x and y to represent a pair of straight lines is

abc + 2fgh – af2 – bg2 –ch2 =0.

Definition of determinant:

To every square matrix A of order n with entries as real or complex numbers, we can associate a number called determinant of matrix A and it is denoted by | A | or det (A) or Δ.

Thus determinant formed by the elements of A is said to be the determinant of matrix A.

If A = `[[a_(11),a_(12)],[a_(21),a_(22)]]` then its | A | =

Minors:

Let | A | = |[aij]| be a determinant of order n. The minor of an arbitrary element aij is the determinant obtained by deleting the ithjth column in which the element aij stands. The minor of aij is denoted by Mij. row and

Cofactors:

The cofactor is a signed minor. The cofactor of aij is denoted by Aij and is defined as

Aij = (− 1)i + j Mij.

Singular and non-singular matrices:

A square matrix A is said to be singular if | A | = 0 (i.e) determinant zero

A square matrix A is said to be non-singular matrix, if | A | ≠ 0. (i.e) determinant is not zero


Examples of determinant zero:

Let us see some examples of determinant zero:

Example 1:

A = `[[1,2, 3],[4,5,6],[7,8,9]]`

Solution:



= 1(45 - 48) - 2(36 -42) +(32 -35)

= -3 +12-9 = 0

|A| = 0

So, A is singular  matrix.


Example 2:

Solve for x if

Solution:

`rArr` (x2 - 35) + (1-2) = 0

`rArr` x2 -35 -1 = 0

`rArr` x2 - 36 =0

`rArr` x2 = 36

`rArr` x = `+-` 6.

These are examples of determinant zero.

Isosceles Acute Triangle

 An Isosceles triangles has

• All angles add up to 180 degrees
• Has two equal sides.
• only one unequal side called the base.
• Base angles of isosceles are all equal.        

An acute triangle is a triangle which has all the angles less than 90°. We know all the internal angles in triangle is equal to 180°. In acute triangle the sum of all the internal angles is equal to 180° but not even the one angle is equal to or greater than 90°.  acute triangle drawn below  L, M and N are all acute angles.

    

Example Problems for Isosceles acute triangle:

Example 1:
   Find the area of the Isosceles triangle of b = 13 cm, h = 7 cm..
Solution:
   Given side of the Isosceles triangle, b = 13 cm, h = 7 cm.
        Area of the Isosceles triangle   = `1/2` * b * h
                                                      = `1/2 ` * 13 * 7
                                                      = 0.5 * 91
                                                      = 45.5 cm²
  Area of  Isosceles triangle   = 45.5 cm²
Example 2:
Find the area of the Isosceles triangle of b = 15 cm, h = 8 cm..
Solution:
Given side of the Isosceles triangle, b = 15 cm, h = 8 cm.
 Area of the Isosceles triangle   = `1/2` * b * h
                                                      = `1/2 ` * 15 * 8
                                                      = 0.5 * 120
                                                      = 60 cm²
  Area of  Isosceles triangle   = 60 cm²
Example 3:
Find the perimeter of Isosceles triangle that has side S₁ =16 cm,S₂ = 16,S₃ = 10 cm.
Solution:
 Given, S₁ =16 cm,S₂ = 16,S₃ = 10 cm.
  perimeter of Isosceles triangle   = S₁+S₂+S₃
                                                         = 16+16+10
  perimeter of Isosceles  triangle  =  42 cm

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Some other examples regarding Isosceles acute triangle:

Example 1:

A triangle has angle 43º, 64º and 73º. What type of triangle is this?

Solution:

  Here all the given angles are less than 90°, hence the given triangle angles are an acute triangle.

Example 2:

A triangle has angle 39º, 65º and 76º. What type of triangle is this?

Solution:

  Here all the given angles are less than 90°, hence the given triangle angles are an acute triangle.

Example 3:

A triangle has angle 45º, 56º and 79º. What type of triangle is this?

Solution:

  Here all the given angles are less than 90°, hence the given triangle angles are an acute triangle.

Associative Property of Addition

Associatively is the property of addition is nothing but in an expression containing two or more occurrences in a row of the addition operator, the order of operations that are performed will not be a matter it can be performed in any order of operation, and rearranging the brackets so that the values will not be changed. Here we are going to see about the associative property of addition

Associative property of Addition is

(a + b) + c = a + (b + c)

Examples for associative property of addition

Example problem 1 for associative property of addition:

Prove that (1 + 2) + 3 = 1+ (2 + 3) by using associative property of addition.

Solution:

Given (1 + 2) + 3 = 1+ (2 + 3) it is in the form (a + b) + c = a + (b + c)

Left hand side:   (1 + 2) + 3 = 3 + 3

= 6

Right hand side: 1 + (2 + 3) = 1 + 5

= 6

Left hand side is equal to right hand side. Therefore it is proved using associative property of addition.
Example problem 2 for associative property of addition:

Prove that (3 + 6) + 9 = 3+ (6 + 9) by using associative property of addition.

Solution:

Given (3 + 6) + 9 = 3+ (6 + 9) it is in the form (a + b) + c = a + (b + c)

Left hand side:   (3 + 6) + 9 = 9 + 9

= 18

Right hand side: 3 + (6 + 9) = 3 + 15

= 18

Left hand side is equal to right hand side. Therefore it is proved using associative property of addition.


Example problem 3 for associative property of addition:

Prove that (2 + 5) + 8 = 2+ (5 + 8) by using associative property of addition.

Solution:

Given (2 + 5) + 8 = 2+ (5 + 8) it is in the form (a + b) + c = a + (b + c)

Left hand side:   (2 + 5) + 8 = 7 + 8

= 15

Right hand side: 2 + (5 + 8) = 2 + 13

= 15

Left hand side is equal to right hand side. Therefore it is proved using associative property of addition.

Nonlinear Equation Algorithms

Nonlinear equation is the form of the multi variable equations and functions. In the equation there will be having more number of terms available. In the nonlinear equations the variables are not dependent to each other in the equation. Nonlinear equation has the more number of different orders of degrees. The graph of the nonlinear equation is not a straight line. It includes the quadratic function of equation and cubic function of equation. Here we are showing about the nonlinear equation algorithms and example problems in it .




Step by step algorithm for solving nonlinear equation:

Nonlinear equation algorithms:

Find whether the given equation is linear or nonlinear.

If the equation is equal to y = m x + b then the equation is linear equation
If the equation is not equal to y = mx + b then the given equation is nonlinear equation.

Find the equations of the order equal to any one of their variable and substitute the value in another equation.
Calculate the value for that equation and find two values.
Substitute the two values of the first equation in the second equation
Now we get each variable having two values.


Nonlinear equation algorithms - Example Problems:

Nonlinear equation algorithms - Problem 1:

Solve the nonlinear equations and find the value of x and y.

x2 - 8y = - 32

- x + y = 4

Solution:

Given equations

x2 - 8y = - 32   ------> Equation 1

- x + y = 4         ------> Equation 2

From the equation 2 rearrange and equal to y

y = 4 + x

Substitute the y value in the equation 1

x2 - 8(4 + x) = - 32

x2 - 32 -8x = -32

x2 - 8x -32+32 = 0

x2 -8x = 0

Solve the above equation by using factorization, we get

x(x - 8) = 0

Therefore the x value will be,

x=0            x - 8 = 0

x = 0 and x = 8

Substitute the x values in equation 2, we get

For x = 0,

y = 4 -x

y = 4 - 0

y = 4

For x = 8

y = 4 - 8

y = - 4

The value of y is 4, - 4


Nonlinear equation algorithms - Practice Problems:

1, Solve the nonlinear equations and find the value of x and y.

x2 - 7y = - 31

- x + y = 3

Answer:

x = 2 , 5 and y = 5, 8

2.

Find the nonlinear equations and find the value of x and y.

x = 2y - 3

y2 + 3x = - 2

Answer:

x = - 1, - 17 and y = 1, - 7

Learn Points and Lines

To learn about points and lines, A point used to represent a place in a plane with a help of pencil, a point is nothing but the dot , it has no dimension or no width, it’s only a simple black dot. In geometry co ordinates of a point which shows the particular place in a segment for representation.Line has two end points is called segment. Line segment is denoted with a connected piece of line.line segments names  has two endpoints and it is named by its endpoints.


learn about points and lines:

To learn about the geometric points and lines we have to know the classification of a points and lines.points and lines classification are as follows.
Collinear points:
When three or more points lies on the same line is said to be collinear points.

Midpoint:
A halfway point where line segment divides into two equal parts are called midpoint.

Equidistant point:
A point which is said to be equidistant in a line segment where point is equal length from other points which are in congruent then the point is equidistant point.

Parallel line segment:
Two lines which does not touch each other are called parallel lines.

Perpendicular line segment:
Two line segment  that form a L shape are called perpendicular lines.


learn problems in points and lines:

Example 1:
Find the distance between the points A(5,2) and B (7,3).


Solution:
Let assume "d" be the distance between A and B.           (x1,y1)= (5,2), (x2,y2)= (7,3).

Then d (A, B) =`sqrt((x2-x1)^2+(y2-y1)^2)`

= `sqrt((7-5)^2 +(3-2))^2)`

= `sqrt(2^2+1^2)`

= `sqrt(4+1)`

=`sqrt5`

Example 2:
Find co-ordinate of the mid point of the line segment joining given points A(-1,1) and B(3,4)

Solution:
The required mid point is
Formul a   `((x_1+x_2)/2 ,(y_1+y_2)/2)` here,  (x1, y1) = (-1,1),(x2, y2) = (3,4)

=  `((-1+3)/(2))``((1 +4)/(2)) `

= `(2/2) ` ,  ` (5/2)`

=`(1,5/2)`

Example 3:
Find the slope of the lines given (2,-1) and (1,3)

Solution:
(x1,y1)= (2,-1), (x2,y2)= (1,3).
We know to find slope of line,m=` (y_2-y_1) /(x_2-x_1)`

=`(3+1)/(1-2)`

m =`4/-1` = -4
Example 4:

Find the equation of the line having slope `1/2` and y-intercept −3.
Solution:
Applying the equation of the line is y = mx + c
Given,       m = `1/2` ,c = −3
y = `1/2` x + (−3)

or  2y = x − 6
or  x− 2y − 6 = 0.

Forms of Quadrilaterals

Forms of Quadrilateral

What is Quadrilateral?

A quadrilateral is a 2-dimensional closed shape with four straight sides.  E.g. The shape ABCD shown here is a quadrilateral.

A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal of the quadrilateral.  AC is a diagonal of quadrilateral ABCD, as is BD.

Properties of a quadrilateral:

Four sides (or edges)
Four vertices (or corners).

The interior angles add up to 360 degrees

Forms of Quadrilateral:

Parallelograms

Quadrilaterals are called parallelograms if both pairs of opposite sides are equal and parallel to each other.  Different parallelograms and their properties are described below.

Opposite sides of a parallelogram are parallel and equal in length.
Opposite angles are equal in size.

Rectangle

Opposite sides of a rectangle are parallel and equal in length.
All angles are equal to 90°



Square

A square has equal sides and every angle is a right angle (90°)
Also opposite sides are parallel.
A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

Rhombus

All sides of a rhombus are equal in length
Opposite sides are parallel.
Opposite angles of a rhombus are equal.

The diagonals of a rhombus bisect each other at right angles

Trapezium

A trapezium has one pair of opposite sides parallel

A regular trapezium has non-parallel sides equal and its base angles are equal, as shown in the diagram

Kite

Two pairs of adjacent sides of a kite are equal in length
One pair of opposite angles (the ones that are between the sides of unequal length) are equal in size.
One diagonal bisects the other.
Diagonals intersect at right angles.

Pyramid of Biomass

Pyramid of Biomass is the graphic representation of biomass of the successive trophic levels of an Eco system. Biomass refers to the total weight of dry matter present in the ecosystem at anyone time. The pyramid of biomass indicates the gradual reduction in biomass at each trophic level from base to top. The total biomass of producers is much more than the total biomass of herbivores. Like wise the total biomass of the third level will be less than the herbivores and so on. The pyramid of biomass are generally upright, but in case of parasites, the pyramid of biomass is inverted. Similarly, if organisms of lower levels are much smaller than those of higher levels, biomass pyramid may be inverted.                                                            

Examples of Pyramid of Biomass.1

Pyramid of Biomass in a Marine ecosystem

In the marine Eco system the Phytoplankton occupy the basic trophic level and the next trophic level is occupied by the Zoo plankton, the third and fourth are occupied by carnivores such as worms, and molluscs and carnivores fishes respectively. Here too the biomass falls in a graded manner from the basic trophic level to higher trophic level.

Examples of Pyramid of Biomass.2

Pyramid of Biomass in  a  grass land 

In a grass land the biomass of grasses is the maximum and it gradually decreases towards the consumer level. That is  the biomass is decrease in this order. Grasses have higher biomass than that of snake  and snake have higher biomass than that of Hawk.

                                                 Grass ------------> Snake -------------> Hawk