Define Matrix Order with Examples? and Types of Matrix Order in Math

What is Matrix Order?

History of Matrix Order

The idea of matrices was introduced by a famous mathematician Arther Cayley in 1857. Matrices are widely used in both physical and social science. In this article we will introduce a new mathematical form, called a matrix, that will enable us to represent a number of different qualities as a single unit.

Details of Matrix:

Before going to know about the Matrix order we can briefly understand matrices. A matrix is a rectangular array of any number or function. Matrices are 2-dimensional .it is represented by rows and columns. Rows are denoted by m and columns are denoted by n. Now we move towards matrix order. The term matrix order is defined as multiply the number of rows and columns in that matrix. We write the order of the matrix in such a manner that the number of rows first show  and then the number of columns.

Types of Matrix order in Maths

What is Matrix order in math?

A matrix is a square or a rectangular array of numbers written within square brackets or parentheses in a definite order, in rows and columns. The term matrix order is defined as the number of rows and columns in that matrix. for example

Generally, the matrices (plural of the matrix) are denoted by capital letters… A, B, C………….etc. while the elements of a matrix are denoted by small letters a, b, c…………. and numbers 1, 2, 3,……..

Easy Matrix Order Example:        

A=\begin{bmatrix}3&5\\5&1\end{bmatrix} ,

B=\begin{bmatrix}2&3&2\\2&2&1\\4&3&4\end{bmatrix} ,

C=\begin{bmatrix}1\\2\end{bmatrix} ,

D=\begin{bmatrix}1&2\end{bmatrix}

We write the order of matrix in such a manner that number of row first show  and then the number of columns

Example

Let suppose a matrix A has m number of row and n number of the column then the order of the matrix is m*n

Let A be matrix

A=\begin{bmatrix}1&2&2\\1&2&0\end{bmatrix}

Here are 2 rows and 3 columns So order of matrix A is 2*3.

Row and column matrix examples

Types of Matrix in Maths

Some important types of matrix order are as follow:

Define Row matrix with Examples

The matrix having only one row is called a row matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&2&3\end{bmatrix}

The order of this matrix is 1×3

Example No 2.

B=\begin{bmatrix}1&2&3&4&5\end{bmatrix}

.Order of this matrix is 1×5 

Define Column matrix with Examples

The matrix having only one column is called a column matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1\\2\\3\end{bmatrix}

Order of this matrix is 3×1

Example No 2.

B=\begin{bmatrix}1\\2\\3\\4\end{bmatrix}

Order of this matrix is 4×1

Define Null Matrix with Examples

A matrix in which all entries are zero is called a null matrix.

Examples:

Example No 1.

A=\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}

Order of this matrix is 3×3

Example No 2.

B=\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}

Order of this matrix is 2×3

Explain Square matrix with Examples

A matrix in which a number of rows and number of columns are equal is called a square matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}

Order of this matrix is 3×3

Example No 2.

B=\begin{bmatrix}1&2\\3&4\end{bmatrix}

Order of this matrix is 2×2

Definition of Diagonal matrix with Easy Examples

A square matrix in which diagonals entries are non-zero and off-diagonals entries are zero is called a diagonals matrix

Examples:

Example No 1.

A=\begin{bmatrix}1&0&0\\0&2&0\\0&0&2\end{bmatrix}

Order of this matrix is 3×3

Example no 2.

B=\begin{bmatrix}1&0\\0&2\end{bmatrix}

Order of this matrix is 2×2

Define Upper triangular matrix with Examples in Math

A square matrix in which all entries below the main diagonals are zero is called upper triangular matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&2&2\\0&2&2\\0&0&5\end{bmatrix}

order of this matrix is 3-by-3

Example No 2.

B=\begin{bmatrix}1&1\\0&1\end{bmatrix}

Order of this matrix is 2-by-2

The upper triangular matrix must be a square matrix.

Lower triangular matrix with Examples

A square matrix in which all items overhead the main diagonals are zero is called a lower triangular matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&0&0\\1&2&0\\2&2&2\end{bmatrix}

Order of this matrix is 3-by-3

B=\begin{bmatrix}1&0&0&\\4&6&0&0\\4&4&5&0\\2&5&4&1\end{bmatrix}

The order of this matrix is 4-by- 4

A lower triangular matrix is also must be a square matrix.

What is Symmetric matrix in Math?

A square matrix in which its transpose is equal to itself is called a symmetric matrix

You can learn more about Matrix in Math on Our Website…

What is Groupoid and Monoid (Semi Group, abelian group)


What is Groupoid in Mathematics?

A groupoid is a non empty set on which a binary operation (steric)* is defined.
in simple words:
A closed set with respect to an operation (steric) is said to be groupoid.

What is Semi Group Explain the Satisfy Condition?


A non empty set S is said to be semi group if it satisfy the following:

  • It it closed w.r.t an operation (steric).
  • The operation (steric) is associative it means it holds associativity.

What is Monoid Explain the Satisfy Condition ?


A semi group or set S is called Monoid if it Satisfy the following:

  • it is closed w.r.t some operation (steric).
  • The operation (steric) holds associativity.
  • It has an identity element.


in simple words:
A semi group which have identity element is used to be said Monoid.

What is Group in Math?


A Monoid which have inverse of each of its elements under the operation (steric) is used to be said group.
in simple manners, a set is said to be a group if it Satisfy the following:

  • G is closed w.r.t some operation (steric).
  • The operation (steric) holds associativity.
  • It has an identity element.
  • every element of G has its inverse in G

How a group become abelian group?


An additional condition is:
⇒ a × b = b × a

it means that if it holds commutativity (commutative property) then G is an abelian group.

Natural Numbers w.r.t to Addition

N={1,2,3,…}
for Addition:
according to first property:

N should be closed i.e., 1+1=2 (2 also in N) It should hold associativity i.e.,

3+(5+8)= (3+5)+8
16=16

Identity exists
as identity element is 0 that doesn’t exist in N


Conclusion
N is a semi_group w.r.t Addition.

         Whole numbers w.r.t to Addition

W={0,1,2,3,…}
i W should be closed i.e.,
3+4=7 (7 is also in W)
ii It should hold associativity i.e.,0+(3+4)=(0+3)+4
7=7
iii additive identity is 0 that also exists in W
7=7
Conclusion
W is a Monoid w.r.t to addition.

    Even numbers w.r.t to Addition

E={2,4,6,…}
i E should be closed i.e.,
4+10=14 (14 also exists in E).
ii It should hold associativity i.e.,
4+(8+10)=(4+8)+10
22=22
iii additive identity is 0 that also exists in E because 0 is an even number.
iv Inverse does not exists in the E.
conclusion
E is Monoid w.r.t Addition.

        Odd numbers w.r.t to Addition

O={1,3,5…}
i O should be closed i.e.,
3+5=8 (8 doesn’t exists in O).
Conclusion
set of odd Numbers is not a group w.r.t addition.

       Prime numbers w.r.t to Addition

p={3,5,7,11,…}
i P should be closed i.e.,
3+7=10 (10 doesn’t exist in P).
ii Associative law holds.
Identity element is 0 that doesn’t exist in P.
Conclusion
set of prime numbers is not a group w.r.t addition.

      Composite numbers w.r.t to Addition

c={4,6,8…}
i C should be closed i.e.,
8+9=17 (17 doesn’t exist in P)
Conclusion
it is also not a group w.r.t addition.

     Rational Numbers w.r.t to Addition

R={0,±1,±2,…}
i R should be closed i.e.,
1+1=2 & -1-1=-2 (closure property exists)
ii It should hold associativity i.e.,
1+(2+3)=(1+2)+3
6=6
iii additive identity that is0 , it also exists in R.
iv inverse is also available in R i.e., additive inverse of 1 is -1.
Conclusion
it is a group w.r.t addition w.r.t Addition.

                  Z^+ w.r.t to Addition

z={±1,±2,±3,…}
i z^+ should be closed i.e.,
1+2=3 (exists in set z^+)
ii It holds associativity.
iii Additive identity that is 0 , doesn’t exist in set z^+.
conclusion
it is a semi_group w.r.t addition w.r.t Addition.

               Z^- w.r.t to Addition

z={-1,-2,-3,…}
i z^- should be closed i.e.,
-1+(-4)=-5 (exists in the set)
ii associativity holds in the set.
iii Additive identity that is 0, doesn’t exist in the set.
Conclusion
set of Z^- integers is semi_group w.r.t Addition.

Matrix multiplication order and its Properties

Matrix multiplication order
Matrix multiplication order

what is Matrix Multiplication Order?

Matrix multiplication Order is a binary operation in which 2 matrices are multiply and produced a new matrix. The new matrix which is produced by 2 matrices is called the resultant matrix. Matrix multiplication is possible only if the number of columns in the first matrix is equal to the number of rows in the second matrix. The resulting matrix is also known as the product matrix. The resulting matrix has the number of rows of the first matrix and the number of columns of the second matrix. The product of two matrices A and B are denoted by AB.

Short History about Matrix Multiplication Order

In 1812 the matrix multiplication order was first described by French mathematician Jacques Philippe Marie Binet to represent the composition of linear maps that are represented by matrices. Matrix multiplication is a basic tool of linear algebra.it is widely used in many fields. Matrix multiplication is applied to Mathematics, statistics, physics, economics, and engineering.

Order of matrix multiplication

Multiplication of two matrices is possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. the resultant matrix has the number of rows in the first matrix and the number of columns in the second matrix. as we know that the order of the matrix is defined as the number of rows multiplied by a number of columns. So the order of matrix multiplication order is that the number of rows of the first matrix and the number of columns of the second matrix.

Example of matrix multiplication order

A=\begin{bmatrix}1&2&3\\1&2&3\end{bmatrix}, B=\begin{bmatrix}1&2&3&4\\2&1&2&3\\3&1&2&3\end{bmatrix}

Let us have two matrices A and B such that the order of matrix A is 2 –by 3 and the order of matrix B is 3 –by 4. the resultant matrix AB is produced and the order of that matrix is 2 –by- 4. Here the 2 is the number of rows in matrix A and 4 is the number of columns in matrix B.

Properties of Matrix Multiplication Order

  • Closure property
  • Commutative property
  • Distributive property
  • Associative property
  • Multiplicative property
  • Identity property of addition
  • Identity property of multiplication

Define Square Matrix with Example

Define Square Matrix with Example

What is Square matrix order?

If a matrix has equal numbers of rows and equal numbers of columns, it is called a square matrix.

Define square matrix with example?

A=\begin{bmatrix}1\end{bmatrix}, B=\begin{bmatrix}1&2\\3&4\end{bmatrix}, C=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}

If a matrix has the same number of rows and columns then it is called a square matrix. Elements in this matrix is arranged in square shape because of the same number of rows and column that’s why we called this matrix square matrix. Rows are shown by m and columns are shown by n. A Square matrice are matrix can be the order of 1 2 3…n. But it is very difficult to work with the order of 11 and above. Basically, a square matrix is denoted by capital words such as E F G B C Z X

                       Matrix m × n

Here the above matrix shows them rows and n column so. As we know that the order of any matrix is multiply by that matrix rows and column so here the order of this matrix is m×n.

Note: the square shape of the matrix is possible if and only if its number of rows is equal to the number of columns such that m=n

How to Square a Matrix 3×3

A=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}

In the above matrix, the number of rows is 3

In the above matrix, the number of columns is 3

The number of rows and columns are equal so it is a square matrix.

The order of this Square matrices is 3 -by- 3

Is 2×3 a Square Matrix?

A=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}

In the above matrix, the number of rows is 2

In the above matrix, the number of columns is 3

The order of the above matrix is 2 -by 3

In the above matrix, the number of rows and number of columns is not equal so it is not a square matrix.

There are two ways to called a square matrix

  • A square matrix of order m or n
  • mTH or nTH order square matrix

4×4 Square Matrix Example

The order of this matrix is 4 -by- 4. we called this matrix:

A=\begin{bmatrix}1&2&3&4\\5&6&7&8\\9&0&1&2\\3&4&5&6\end{bmatrix}
  1. a Square matrices of order 4
  2. 4 order Square matrices

Square of Matrix 2×2

B=\begin{bmatrix}1&2\\3&4\end{bmatrix}

The order of this matrix is 2 -by 2. We call this matrix as

  1. A Square matrices of order 2
  2. 2 order Square matrices

Can a Square matrix be 1×1?

Order of this matrix is 1-by-1. We call this matrix as:

A=\begin{bmatrix}1\end{bmatrix}
  1. A Square matrices of order 1
  2. 1 order Square matrices

Scalar and Vector Quantity examples

In mathematics, we encounter two important quantities known as scalar and vector

Scalar Quantity:

Scalar is one that possesses the only magnitude. it can be specified by a number along with a unit. The quantities like mass, time, density, temperature, length, volume, speed, and work are examples of scalar.

Vector Quantity:

Vector is one that possesses both magnitude and direction. The quantities like displacement, velocity, acceleration, weight, force, momentum, electric, and magnetic field are examples of vectors.

Now here we discuss the vector and their fundamental operation. We begin with a geometric interpretation of vectors in plane and space.

Geometric Interpretation Of Vector:

A geometric vector is represented by a directed line segment AB with A is in its initial point and B in its terminal point. It is often found convention to denote a vector by an arrow and is written either as \overrightarrow{AB} or as a boldface symbol like v or in underline Form \underline v.

Magnitude of a vector

It is also called a length or norm of a vector. The magnitude of a vector \overrightarrow{AB} or \underline v is its absolute value and is written as \left|\overrightarrow{AB}\right| or simply AB or \left|\underline v\right|.

Unit Vector:

A unit vector is defined as a vector whose magnitude is unity. The unit vector of vector \underline v is written as \left|\widehat{\underline v}\right| (read as \underline v hat) and is defined by \widehat{\underline v}\;=\frac{\underline v}{\left|\underline v\right|}.

Zero Or null Vector :

If terminal point B of a vector \overrightarrow{AB} coincides with its initial point A. Then magnitude AB = 0 and \overrightarrow{AB} = \underline0 which is called zero or null vector.

Negative Vector:

two vectors are said to be negative of each other

if they have same magnitude but opposite direction

if \overrightarrow{AB\;\;}=\underline v then \overrightarrow{BA\;\;}=\overrightarrow{-AB}\;=-\underline{\;v}

and \left|\overrightarrow{BA}\right|\;=\;\left|\overrightarrow{-AB}\right|

Multiplication of vector by a scalar:

as we know that the word scalar mean to a real number. multiplication of a vector \underline v by a scalar k is a vector whose magnitude is k times of that of \underline v .it is denoted by k \underline v .

if k is positive then \underline v and k \underline v are in the same direction.

if k is negative then \underline v and k \underline v are in opposite direction.

Equal vector

Define Convergence of Power Series with all Test

Define Convergence of Power Series?

An infinite series of the form

c_0 + c_1x + c_2x^2 + … + c_{n\;}x^n + … = \overset\infty{\underset0{\sum c_n}}x^n = 0 (1)

Is called a power series in x. the coefficient c_n are real number and x is a real variable.

Define Convergence of Power Series with all Test

A series of the form

c_0 + c_1\left(x-a\right) + c_2\left(x-a\right)^2 + … +c_n\left(x-a\right)^n + … = c_n\left(x- a\right)^n (2)

is a power series in \left(x-a\right) where a is a real number. However the series (2) can always be reduced to a series of the form (1) by putting \left(x-a\right)\;=y . But here we shall study the power series of form (1).

What is Convergence of Power Series?

If a numerical value of X is substituted into (1) then it is a series of the constant terms.

Let \left\{S_n\right\} be the sequence of partial sum of the series \overset\infty{\underset1{\sum a_n}}. If the sequence \left\{S_n\right\} converges to the limit S, then the \overset\infty{\underset1{\sum a_n}} is said to converge and S is called the sum of the series. In this case we write

\overset\infty{\underset1{\sum a_n}}\;=\;S

If the sequence \left\{S_n\right\} diverges then the series \overset\infty{\underset1{\sum a_n}} is said to be divergent.

Convergence of Power Series test

There are some tests to check whether the series is converged is diverge

  • The Basic Comparison Test
  • The Limit Comparison Test
  • The Integral Test
  • The Ratio Test
  • Cauchy’s Root Test

Define Basic Comparison Test?

Let \overset\infty{\underset1{\sum a_n}} and \overset\infty{\underset1{\sum b_n}} be series of positive terms with a_{n\;}\;\leq\;b_n for each n=1,2,3…. then

  • If \overset\infty{\underset1{\sum b_n}} converges, then \overset\infty{\underset1{\sum a_n}} is converges.
  • If \overset\infty{\underset1{\sum a_n}} diverges then the \overset\infty{\underset1{\sum b_n}} is diverge.

What is Limit Comparison Test?

Let \overset\infty{\underset1{\sum a_n}} and \overset\infty{\underset1{\sum b_n}} be series of positive terms. Then

  • If \lim_{n\rightarrow\infty}\frac{a_n}{b_n}\;=\;L\;\neq0, then either both series converge or both diverge.
  • If \lim_{n\rightarrow\infty}\frac{a_n}{b_n}\;=\;0 and \overset\infty{\underset1{\sum b_n}} converge then \overset\infty{\underset1{\sum a_n}} is also converges.
  • If \lim_{n\rightarrow\infty}\frac{a_n}{b_n}\;=\;\infty and \overset\infty{\underset1{\sum b_n}} diverges then \overset\infty{\underset1{\sum a_n}} is also diverges.

What is Integral Test?

Let \overset\infty{\underset1{\sum a_n}} be a positive term series.if f is the continous and nonincreasing function on [ 1, \infty ) such that f (n) = a_n for all positive integer n, then

  • \overset\infty{\underset1{\sum a_n}} convergers if \int_1^\infty f\left(x\right)\operatorname dx converges.
  • \overset\infty{\underset1{\sum a_n}} diverge if \int_1^\infty f\left(x\right)\operatorname dx diverges.

What is Ratio Test?

let \overset\infty{\underset1{\sum a_n}} be a serie sof positive terms and suppose tha \lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\;=\;L , where L is a non negative real number.

  • If L< 1, the series \overset\infty{\underset1{\sum a_n}} converges.
  • If L> 1 ,or infinity the series \overset\infty{\underset1{\sum a_n}} converges.
  • If L= 1 the test fails to determine converges or diverges of the series.

What is Cauchy’s Root Test?

Let \overset\infty{\underset1{\sum a_n}} be a series of positive terms and suppose that \;\lim_{n\rightarrow\infty}\left(a_n\right)^{1/n}\;=L where L is a nonnegative real number or infinity.

  • If L< 1, the series [katex]\overset\infty{\underset1{\sum a_n}}[/katex] converges.
  • If L> 1 ,or infinity the series \overset\infty{\underset1{\sum a_n}} converges.
  • If L=1 the test fails.

Find the Interval of Convergence of the Power Series

The set of all values of X for which a power series converges  is called the interval of convergences

Define Radius of Convergence of Power Series

The number R in any interval is called the radius of convergence of the series.

How do you find the Radius of Convergence of Series?

Every power series has an interval of convergence and radius of convergence by the power series \sum_{n=0}^\infty\;c_nx^n converge the only for x= 0 then its interval of convergence is reduced to the point 0 and its radius of convergence is 0.

If the power series for all values of x, then its interval of convergence Is interval \left(-\infty\;\infty\right) and its the radius of convergence is Infinity.

If the power series converges for \left|x\right|\;<\;R[/katex] and diverges for [katex]\left|x\right|\;>\;R then the radius of converges is R ad its interval is one of the interval \left(-R,\;R\right) or \left[-R,\;R\right].

Addition and Multiplication of Power Series with Equation

Let \sum_{n=0}^\infty\;a_nx^n and \sum_{n=0}^\infty\;b_nx^n be two power series with a common interval of convergence.then

  • \sum_{n=0}^\infty\;a_nx^n + \sum_{n=0}^\infty\;b_nx^n = \sum_{n=0}^\infty\left(a_n+x_n\right)x^n
  • (\sum_{n=0}^\infty\;a_nx^n) (\sum_{n=0}^\infty\;b_nx^n ) = \sum_{n=0}^\infty\;c_nx^n

Where c_{n\;=\;\;}a_0b_n+a_1b_{n-1}+…+a_{n-1}b_1+a_nb_0 power series may be added and multiplied together much the same way as a polynomial.

  A power series may also be divided by another power series in a manner similar to the division of polynomials.

The interval of convergence of each of the new power series obtain by the algebraical operation is a common interval of convergence of the two given power series.

Differentiation and Integration of Power series

Each power series \sum_{n=0}^\infty\;c_nx^n defines a function f where

f(x) = \sum_{n=0}^\infty\;c_nx^n

for each X in the interval of convergence of the power series. \sum_{n=0}^\infty\;c_nx^n is called a power series representation of f(x). The function f is continuous and differentiable. Taylor and Maclaurin’s formula is used to obtain a power series representation of a function having derivatives of any order on the same interval. It is known that the geometric series 1+x+x^2+… converges for -1 < x< 1 and its sum is[katex]\frac1{1-x}[/katex]. thus a power series representation of [katex]\frac1{1-x}[/katex] is the power series [katex]1+x+x^2+…[/katex] whose radius of convergence is 1.

Having obtained a function f representative by power series. A natural question arises whether their function can be differentiated and integrated. The answer is that the new power series can be obtained by term-by-term differentiation and integration of a given power series within its interval of convergence.

Convergence and Divergence of Infinite Series

Series:

The sum of an indicated number of terms in a sequence is called a series. Series is widely used in calculus and mathematical analysis. Series are also used in many fields like physics statistics finance etc.

History of series:

  • Archimedes  a Greek mathematician produce the first  known summation of an infinite Series with a method that is used in the field of calculus today.He used the method of exhaustion to find the area under the arc of a parabola with the summation of infinite series.
  • In 17th century the Mathematician James Gregory work in new decimal system om infinite series and published many Maclaurine series.
  • In 1715 the mathematician Brook Taylor gave a general method for constructing the Taylor series for all function for which they exist.

Example:

The sum of the first seven terms of the sequence {n²} is the series

1+4+9+16+25+36+49.

The above series is also named as the 7th partial sum of the sequence {n²}.

If the number of terms in a series is finite then the series is called a finite series while a series consisting of an unlimited number of terms is an infinite series.

Notation:

The Greek letter E is used to represent the sum of series. This notation is called summation or sigma. For example, the series 3+6+9+12+15 can be represented as 3\overset\infty{\underset{n=1}{\sum a_n}} where the n is called the index of summation.

Sum of first n terms of an Arithmetic series:

for any sequence {a_n} we have

S_n = a_1 + a_2 +a_3 +….+ a_n

if {an} is an A.P then S_n can be written with usual notation as

S_n = a_1 +( a_1 +d) + ( a_1 + 2d)+ …. +( a_n -2d) +( a_n -d) + a_n (1)

if we write the terms of series in the reverse order the sum of nterms remain the same that is

S_n = a_n +( a_n -d) = ( a_n -2d)+…….( a_1 +2d) = (a_1 +d) + a_1 (2)

adding (1) and (2) we get

2 S_n = ( a_1 + a_n ) +( a_1 + a_n ) +( a_1 + a_n ) =….=( a_1 + a_n ) +( a_1 + a_n ) +( a_1 + a_n )

S_n = ( a_1 + a_n ) +( a_1 + a_n )….to n terms

n( a_1 + a_n )

thus S_n = \frac n2 ( a_1 + a_n )

S_n =\frac n2[ a_1 + a_1 +(n-1)d]

S_n = \frac n2 [2 a_1 +(n-1)d]

Infinite Series:

Let {an} be a sequence .an expression of the form

a_1 + a_2 +a_3 +….+ a_n +….

containing infinitely many terms of the sequence of {an} is called an infinite series.

this series is symbolically written as.

\overset\infty{\underset{n=1}{\sum a_n}} or \overset\infty{\underset1{\sum a_n}}

here the a_n is called the nth term of the series.

Let S_n denote the sum of the first n terms of this series then we write

S_1 = a_1

S_2 = a_1 + a_2

S_3 = a_1 + a_2 + a_3

.

.

.

S_n = a_1 + a_2 + a_3 +……+ a_n

the sequence { S_n } is called the sequence of partial sums of the upper series and the number S_n is called the nth partial sum of the series \overset\infty{\underset1{\sum a_n}}

Converge and Diverge:

Let { S_n } be the sequence of partial sum of the series \overset\infty{\underset1{\sum a_n}} . if the sequence { S_n } converge to the limit s, then \overset\infty{\underset1{\sum a_n}} is said to be converge and S is called the sum of series .

if the sequence diverge then the series is said to be divergent.

To check the convergence of series we may have use some tests. These test are tell us about the convergence or divergence of series. These tests are following.

  • The Basic Comparision Test
  • The Limit Comparision Test
  • The Integral Test
  • The Ratio Test
  • The Cauchy Test

Upper and Lower Triangular Matrix

Triangular Matrix:

A square matrix in which all the entries upper or below the maim diagonals elements are zero is said to be a triangular matrix. The triangular matrix must be a square matrix that means the triangular matrix has the same number of rows and columns.

There are two types of triangular matrices. the upper triangular matrix and lower triangular matrix

Types of a Triangular Matrix:

Upper Triangular Matrix:

Upper triangular matrix

A square matrix in which entries below the main diagonals are zero is called an upper triangular matrix.

For Example:

\begin{bmatrix}5&3&2&5\\0&7&7&9\\0&0&4&1\\0&0&0&8\end{bmatrix}

Lower Triangular Matrix:

lower triangular matrix

A square matrix in which the entries above the main diagonals are zero is called a lower triangular matrix.

For Examples:

\begin{bmatrix}9&0&0&0\\6&2&0&0\\7&3&8&0\\3&5&9&6\end{bmatrix} \begin{bmatrix}6&0&0\\9&1&0\\7&3&7\end{bmatrix}

Notation

The capital word U is the notation of the upper triangular matrix it is also called the right triangular matrix. And the capital word L is used for the lower triangular matrix it is also called the left triangular matrix.

Properties of an Upper Triangular Matrix:

  • If we added or subtract two upper triangular matrices then the matrix that we obtain is also a upper triangular matrix
  • If we multiply two upper triangular matrix the resultant matrix also upper triangular matrix
  • If we take inverse of upper triangular matrix then the resultant matrix is upper triangular matrix
  • Transpose  of upper triangular matrix is lower triangular matrix such that U^t = L.
  • The upper triangular remain unchanged when we multiply it by any scalar quantity .

Properties of a Lower Triangular Matrix:

  • If we added or subtract two lower triangular matrices then the resultant matrix is a lower triangular matrix.
  • If we multiply two lower triangular matrix the resultant matrix also lower triangular matrix
  • If we take inverse of lower triangular matrix then the resultant matrix is lower triangular matrix
  • Transpose  of lower triangular matrix is lower triangular matrix such that L^t</span> = U .
  • The lower triangular remain unchanged when we multiply it by any scalar quantity .

Some other types of a triangular matrix

Unit Triangular Matrix:

In any upper or lower triangular matrix if the main diagonals entries are 1 then it is called unit (upper or lower) triangular matrix. Mostly it is called normed triangular matrix

\begin{bmatrix}1&0&0&0\\6&1&0&0\\7&3&1&0\\3&5&9&1\end{bmatrix}\; lower unit triangular matrix

\begin{bmatrix}1&6&7&8\\0&1&7&3\\0&0&1&6\\0&0&0&1\end{bmatrix}\; upper unit triangular matrix

Strictly Triangular Matrix:

In any upper or lower triangular matrix if the elements of the main diagonal are zero then it is called strictly upper or lower triangular matrix

\begin{bmatrix}0&6&7&8\\0&0&7&3\\0&0&0&6\\0&0&0&0\end{bmatrix}\; upper strictly triangular matrix.

\begin{bmatrix}0&0&0&0\\5&0&0&0\\6&1&0&0\\9&7&5&0\end{bmatrix}\; lower strictly triangular matrix.

Properties of Inverse Matrix Proof

What is Inverse Matrix?

Let A be a square matrix of order n. A matrix B of order n is said to be the inverse of A if AB=BA=I_n

Note:

The inverse of an arbitrary square matrix may not exist.

Example of Inverse Matrix ?

The Matrix A=\;\begin{bmatrix}1&-1\\-1&1\end{bmatrix} has no inverse.

An inverse of a matrix, if it exists, is unique. For suppose that A has two inverses, say B and C. Then

AB = BA=I and AC = CA= I

Therefore, by the associative law for multiplication, we have

B(AC) = BI = B

and (BA)C = IC = C

and so B = C

The unique inverse of a matrix A, if it exists, is denoted by A^{-1}

A square matrix A, whose inverse exists, is called a nonsingular or invertible matrix. Square matrices which do not have inverse are called singular matrices.

Properties of Inverse Matrix Proof

It is easy to verify the following properties of inverses of matrices:

  • For any invertible matrix A,(A^{-1})^{-1} = A
  • For nonsingular matrices A and B, (AB)^{-1} = B^{-1} A^{-1}
  • For any invertible matrix A, (A^t)^{-1} = (A^{-1})^t

Inverse Matrix Proof

For nonsingular matrices A and B, (AB)^{-1} = B^{-1} A^{-1}.

Since A and B are nonsingular, A^{-1} and B^{-1} exist. Also, Since A and B square matrices AB are defined.

To prove that AB is nonsingular and B^{-1} A^{-1} is the inverse of AB, We show that

(AB)

Diagonal Matrix Examples

What is Diagonal Matrix?

In mathematics, the term diagonals matrix define as the matrix in which the off diagonals entries are zero and main diagonals entries are some else. There is no restriction for main diagonals entries. diagonals matrix is also called the scaling matrix because when we multiply any matrix with diagonals matrix it change the scale of that matrix here the scale meaning size of the matrix. The identity matrix is one of the examples of the diagonals matrix. because in the identity matrix the main diagonals are 1 and off diagonals are zero.

Diagonal Matrix Definition?

Diagonals matrix must be a square matrix. Since the square matrix is a symmetric matrix so we also called this matrix the symmetric diagonals matrix.

Examples of diagonals matrix

Example no 1:
\begin{vmatrix}6&0\\0&4\end{vmatrix}

It is a 2 by 2 diagonals matrix.

Example no 2:
\;\begin{bmatrix}7&0&0\\0&5&0\\0&0&2\end{bmatrix}

It is 3 by 3 diagonals matrix.

Example no 3:
\begin{bmatrix}7&0&0&0\\0&5&0&0\\0&0&4&0\\0&0&0&9\end{bmatrix}

it is a 4 by 4 diagonals matrix.

Properties of Diagonals matrix

  • When we add two diagonals matrices then the resultant matrix also diagonals matrix. \;\begin{bmatrix}1&0&0\\0&4&0\\0&0&3\end{bmatrix}\;\;+\begin{bmatrix}7&0&0\\0&5&0\\0&0&2\end{bmatrix}\;=\begin{bmatrix}8&0&0\\0&9&0\\0&0&5\end{bmatrix}
  • If any matrix is triangular and normal then then it is diagonal matrix.
  • both the upper and lower triangular matrix are diagonals matrix.
  • Symmetric matrix also diagonals matrix.
  • When we multiply any matric with diagonals matric then the matrix which we obtain is diagonals matrix \;\begin{bmatrix}2&0&0\\0&4&0\\0&0&5\end{bmatrix}\;\;\times\;\;\begin{bmatrix}4&0&0\\0&3&0\\0&0&2\end{bmatrix}\;=\begin{bmatrix}8&0&0\\0&12&0\\0&0&10\end{bmatrix}
  • Zero matrix is diagonals matrix. \;\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\; it is a 3 by 3 zero matrix and it hold the property of diagonal matrix.
  • Identity matrix also diagonals matrix \;\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} this is the 3 by 3 identity matrix since it is hold the property of diagonals matrix
  • The matrix of order 1 by 1 is diagonals matrix.
  • Transpose of any diagonals matrix is same as matrix.
  • Inverse of diagonals matrix is resiprocal of main diagonals entries .
  • in under multiplication the diagonals matrix are commutative as AB = BA

Note Diagonal Matrix

  • since 1 by 1 order matrix is also diagonal matrix.
  • there is no restrication for diagonal entries.