Defination of Matrix

Definition of a Matrix:

A rectangular array of entries is called a Matrix. The entries may be real, complex or functions.

The entries are also called as the elements of the matrix.

The rectangular array of entries are enclosed in an ordinary bracket or in square bracket. Matrices are denoted by capital letters.

Example:

(i)

Note that the entries in a given matrix need not be distinct.

(ii)

The entries in this matrix are function of x.

A matrix having m rows and n columns is called as matrix of order mxn. Such a matrix has mn elements.

In general, an mxn matrix is in the form

Where a_ij represents the element in i^th column.

The above matrix may be denoted as A = [a_ij]_mxn.

Introduction to algebra with integers:

Introduction to algebra with integers:

The set of integers, Z, consists of the whole numbers and their negative counterparts. Z = { …, -3, -2, -1, 0, 1, 2, 3, … }
The absolute value of a number is the distance between the number and zero on a number
It is defined by the formula: x = x, if x ≥ 0 − x, if x <>
The algebra with integers is the set of integers that has whole number and their negative counterparts. The algebra with integers include different operations as addition, subtraction, multiplication, division of algebra with integers.Let us see the algebra with integers concepts and example problems.

Types of Integers

Types of Integers:

There are two types of Integers:

1. Positive Integers:
Positive integers are whole numbers, which are greater than zero. For example, 25, 27, 103, 758…etc.
2. Negative Integers:
Negative integers are the opposites of the whole numbers. For example, -5, -22, -38, -504, -4585…etc. Negative numbers indicated by the sign (-). Zero is neither positive nor negative.

Rules for Dividing Integers:

The rules for solving dividing integers is explained below:
Consider this division example: 24 ÷ 4 = 6.

In division each number is referred by a special name.

Here, 24 is dividend, 4 is divisor, and 6 is quotient

quotient × divisor = dividend

dividend ÷ divisor = quotient

dividend ÷ quotient = divisor

Rules for Solving Dividing Integers

1) Positive ÷ Positive = Positive

Example: 28 ÷ 7 = 4

28, 7, and 4 are positive.

2) Negative ÷ Negative = Positive

Example: -28 ÷ -7 = 4

28 and 7 are negative, but 4 is positive.

3) Negative ÷ Positive = Negative

Example: -28 ÷ 7 = -4

28 is negative, 7 is positive, but 4 is negative.

4) Positive ÷ Negative = Negative
Example: 28 ÷ -7 = -4

28 is positive, 7 is negative, and 4 is negative.

Geometrical Interpretation - Scalar Triple Product Proof

Suppose there exists a parallelepiped with vectors a, b and c along sides OA, OB and OC respectively.

Height = OA

= a cos

where angle which the height OA makes with the base of the parallelepiped is

Area of base = area of parallelogram OBDC

= | b * c | (from definition of cross product)

= |b| |c| sin
where angle between OB and OC is
Volume of parallelepiped = Area of base * Height
= Area of parallelogram OBDC * OA
= (|b| |c| sin theta) * ( |a| cos alpha)
= ( |b| * |c| ) a cos= a . ( b * c)

Definition- Calculate Ratio Math

Definition- Calculate Ratio Math:

The ration of two numbers r and s(s≠0) is the section of the numbers. The numbers r and s are called the terms of the ratio.

Concept - calculate ratio math:

The numeric ratio of two numbers r and s(s≠0) is the section of the numbers. The numbers r and s are called the conditions of the numeric ratio.

Types of ratio- calculate ratio math:

1. Compounded ratio in math.
2. Duplicate ratio in math.
3. Triplicate ratio in math.

Perfect square of a trinomial

Perfect square of a trinomial:

If all the terms of the polynomial have a common factor, we take out the common factor and factorise.
If the polynomial can be expressed as the difference of two squares,
we use a² - b² = (a + b) (a - b).

• If all the terms of the polynomial have a common factor, we take out the common factor and factorise .
• If all the terms of the polynomial have a common factor, we take out the common factor and factorise .

• If the polynomial can be expressed as the difference of two squares,

we use a² - b² = (a + b) (a - b)

• Quadratic trinomials of the form x² + ax + b can be factorised using the identity. (x + a) (x + b) = x² + x(a + b) + ab.

• When the trinomial is ax² + bx + c and , we follow the following steps. We find two factors whose sum is b, and whose product is a x c.

We split the middle term using these two factors and factorise by grouping the terms.

• If the polynomial can be expressed as the sum or difference of two cubes we use the following identities.

a³ + b³ = (a + b) (a² - ab + b²)
a³ - b³ = (a - b) (a² + ab + b²)

What are equal and parallel line?

What are equal and parallel line?


If two parallel lines are cut by a transverse, the alternate angles are equal.

and
These are two pairs of alternate angles.



A transversal intersects two lines. If the alternate angles are equal, then the lines are parallel.
If or then AB is parallel to CD.