## Sunday, May 8, 2016

### XII - 11.24 Solution of a homogeneous system of linear equations - Video Lectures

Solving a Homogeneous System

NightingaleMath

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### XII - 11.23 Rank method - Video Lectures

9 May

A number r is said to be the rank of a an "m x n" marix if

i) every square sub matrix of it of order (r+1) or more is singular, and

ii) there exists at least on square matrix of order r which is non-singular.

In other words, the rank of a m x n matrix is the order of the highest order non-singular square submatrix of it.

Solving 4x4 Linear Equations by Rank of Matrix Method_Detailed Step by Step Explanation

Sujoy Krishna Das

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A number r is said to be the rank of a an "m x n" marix if

i) every square sub matrix of it of order (r+1) or more is singular, and

ii) there exists at least on square matrix of order r which is non-singular.

In other words, the rank of a m x n matrix is the order of the highest order non-singular square submatrix of it.

Solving 4x4 Linear Equations by Rank of Matrix Method_Detailed Step by Step Explanation

Sujoy Krishna Das

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### XII - 11.22 Solution of a non-homogeneous system of linear equations - Video Lectures

9 May

Non-Homogeneous system of equation with infinite solution

Rahul Abhang

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Nonhomogeneous System Solutions

TheTrevTutor

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Non-Homogeneous system of equation with infinite solution

Rahul Abhang

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Nonhomogeneous System Solutions

TheTrevTutor

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### XII - 11.21 System of simultaneous linear equations - Video Lectures

Consistency of a System of Linear Equations

MathDoctorBob

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Consistent And Inconsistent System of Equations Example - 1 / Matrices / Maths Algebra

We Teach Academy Maths

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### XI - 2.4 Some results on relations - Video Lectures

1. If R and S are two equivalence relations on a set A, then R∩S is also an equivalence relation on A.

2. The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.

3. If R is an equivalence relation on a set A, the R

Proof of Set operations in Relations -1 / NCERT Std XI Mathematics

MathsMynd

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2. The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.

3. If R is an equivalence relation on a set A, the R

^{-1}is also an equivalence relation on A.Proof of Set operations in Relations -1 / NCERT Std XI Mathematics

MathsMynd

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### XI - 2.5 Composition of relations - Video Lectures

When r and S are two relations from set A to B and B to C respectively, we can define a relation SoR from A to C such that

(a.c) Є SoR imples for all b Є B subject to the relations (a,b) ЄR and (b.c) ЄS.

SoR is called the composition of R and S.

Properties of SoR

In general RoS is not equal to SoR.

(SoR)

^{-}= R

^{-}oS

^{-}

^{}

Composition of relations

Math 290, GMU

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### XII - 11.20 Echelon form of a matrix - Video Lectures

Elementary Linear Algebra: Echelon Form of a Matrix, Part 1

James Hamblin

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### XII - 11.19 Equivalent matrices - Video Lectures

Operations that Produce Row Equivalent Matrices

psccmath

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### XI - 3.6 Function as a relation - Video Lectures

Maths Relation and Functions Part 1 (Relation function concept) Mathematics CBSE Class X1

ExamFearVideos

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### XI - 3.5 Number of functions - Video Lectures

Number of functions

Gate Lectures by Ravindrababu Ravula

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### XI - 3.4 Equal function - Video Lectures

Example On Equal Functions / Maths Algebra

We Teach Academy Maths

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### XI - 3.2 Domain, Co-Domain and range of a function - Video Lectures

Domain, Codomain, and Range

Worldwide Center of Mathematics

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### XI - 3.1 Function - Video Lectures

Functions, Lecture 4 , Maths IIT JEE ( Definition of functions)

Collegepedia.in

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### XI - 3.8 Composition of functions - Video Lectures

Composition of Functions

ProfRobBob

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### XI - 3.10 Inverse of an element - Video Lectures

Inverse elements for Binary operations : ExamSolutions Maths Revision

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Examsolutions

### XI - 3.11 Inverse of a function - Video Lectures

Inverse Functions

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ProfRobBob

### XI - 3.12 Properties of inverse of a function - Video Lectures

15 May

Properties of the Inverse Image of a Function on Sets: Practice With Proof

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Properties of the Inverse Image of a Function on Sets: Practice With Proof

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## Thursday, May 5, 2016

### XI - 2.3 Types of relations - Video Lectures

XI -

2.3 Types of relations - Video Lectures

2.3 Types of relations

Void relation

Universal relation

Identity relation

Reflexive relation

Symmetric relation

Transitive relation

Antisymmetric relation

Equivalence relation

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We teach academy mathematics

### XI - 2.2 Relations - Video Lectures

XI - 2.2 Relations - Video Lectures

2.2 Relation

Let A and B be two sets. Then a relation R from A to B is a subset of A×B.

R is a relation from A to B => R is a subset of A×B.

Total number of relations: If A and B are two non empty sets with m and n elements respectively, A×B consists of mn ordered pairs.

Since each subset defines a relation from A to B, so total number of relations from A to B is 2

^{mn}.

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We teach academy mathematics

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Kadas Learning

### XI - 2.1 Cartesian product of sets - Video Lectures

XI - 2.1 Cartesian product of sets - Video Lectures

2.1 Cartesian product of sets

Cartesian product is an operation on sets.

Ordered pair: An Ordered pair consists of two objects or elements in a given

**fixed order.**

Cartesian product: Let A and B be any two non empty sets. The set of all ordered pairs (a,b) such that a ЄA and b ЄB is called the Cartesian product of the sets A and B and is denoted by A×B

Theorems

Theorem 1; For any three sets

(i) A×(B U C) = (A×B) U (A×C)

(ii) A×(B∩C) = (A×B) ∩(A×C)

Theorem 2: For any three sets

A×(B – C) = (A×B) – (A×C)

Theorem 3: If and A and B are any two non-empty sets, then

A×B = B×A => A = B

Theorem 4: If A is a subset of B, A×A is a sub set of (A×B) ∩(B×A)

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IMA

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IMA

### XII - 11.14 Elementary transformations of Elementary Operations of a matrix - Video Lectures

XII -

11.14 Elementary transformations of Elementary Operations of a matrix - Video Lectures

1. Interchange of two rows or columns.

2. Multiplication of all elements of a row or column of a matrix by a non-zero scalar,

3. Addition to the elements of a row or column of the corresponding elements of any other row (to a row) or any other column (to a column) multiplied by a scalar k.

**Elementary matrix:**A matrix obtained from an identity matrix by a single elementary operation (transformation) is called an elementary matrix.

### Elementary Operation of matrix - all three operations - Video

https://www.youtube.com/watch?v=1k7-qh3mj4k

## Finding Inverse of a Matrix Using Elementary Transformations

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maths1122

### XII - 11.13 Inverse of a matrix - Video Lectures

Let A be a square matrix of order n

If AB = I

_{n}= BA

The B is inverse of A and is written as

A

^{-1}= B

**Theorems related to Inverses of matrices**

1. Every invertible matrix possesses a unique inverse

2. A square matrix is invertible iff it is nonsingular.

3. A

^{-1}= (1/|A|)adj A

4. Cancellation laws: Let A, B, and C be square matrices of the same order n. If A is a non-singular matrix, then

(i) AB = AC => B = C … (left cancellation law)

(ii) BA = CA => B = C … (right cancellation law)

This law is true only when |A| ≠ 0. Otherwise, there may be matrices such that AB = AC but B≠C.

5. Reversal law: If A and B are invertible matrices of the same order, then AB is invertible and

(AB)

^{-1}= B

^{-1}A

^{-1}

6.If A,B,C are invertible matrices then

(ABC)

^{-1}= C

^{-1}B

^{-1}A

^{-1}

7.If A is an invertible square matrix, then A

^{T}is also invertible and

(A

^{T})

^{-1}= (A

^{-1})

^{T}

8. Let A be a non-singular square matrix of order n. Then

|adj A| = |A|

^{n-1}

9. If A and B are non-singular square matrices of the same order, then

adj AB = (adj B) (adj A)

10. If A is an invertible square matrix, then

adj A

^{T}= (adj A)

^{T}

11. If A is a non-singular square matrix, then

adj(adj A) = |A|

^{n-2}A

## Inverse of 2x2 matrix

Math Meeting__________________

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Inverse of 3x3 matrix

Math Meeting

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Matrix Inverse Properties

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slcmath@pc

## Tuesday, May 3, 2016

### XII - 11.12 Adjoint of a matrix - Video Lectures

XII - 11.12 Adjoint of a matrix - Video Lectures

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Exam Fear Videos

Adjoint of matrix order 2X2

FreeTutorialsWorld

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Adjoint of a 3x3 matrix

Astryl

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### XII - 11.11 Singular matrix - Video Lectures

XII - 11.11 Singular matrix - Video Lectures

A square matrix is a singular matrix if its determinant is zero

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KhanAcademy

### XII - 11.9 Symmetric and skew symmetric matrices - Video Lectures

XII -

11.9 Symmetric and skew symmetric matrices - Video Lectures

Symmetric matrix

A square matrix is called a symmetric matrix iff a

_{ij}= a

_{ji}for all I,j.

It means (A)

_{ij}= (A

^{T})

_{ij}

skew symmetric matrix

A square matrix is called a skew-symmetric matrix iff a

_{ij}= -a

_{ji}for all I,j.

It means (A)

_{ij}= -(A

^{T})

_{ij}

It means A

^{T}= -A

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Techtud

Problem on 11.9 Symmetric and skew symmetric matrices - Video Lectures

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Khanacademy

### XI - 1.11 Some important results on number of elements in sets - Video Lectures

XI -

1.11 Some important results on number of elements in sets - Video Lectures

## Finding the Number of Elements in a Set

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MATH 110

## Important Results on Number of Elements on Sets

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AppuSeriesAcademy

## Monday, May 2, 2016

### XII - 11.8 Transpose of a matrix - Video Lectures

XII -

11.8 Transpose of a matrix - Video Lectures

Tranpose of a matrix A

^{T}is obtained from A by changing its rows into columns and its columns into rows.

The first row of A is the first column of A

^{T}.

Properties of Transpose

1. (A

^{T})

^{T}= A

2. (A+B)

^{T}= A

^{T}+B

^{T}( A and B must have the same order)

3. (kA)

^{T}= kA

^{T}., (k is any scalar)

4. (AB)

^{T}= B

^{T}A

^{T}

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https://www.youtube.com/watch?v=uZYIZ5M2DaU

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Example Problem

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Ram Polepeddi

### XII - 11.6 Subtraction of Matrices - Video Lectures

Class XII - Chapter Matrices

11.6 Subtraction of Matrices - Video Lectures

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numericalmethodsguy

### XII - 11.7 Multiplication of matrices - Video Lectures

Class XII - Chapter Matrices

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ProfRobBob

### XII - 11.5 Multiplication of a matrix by a scalar - Video Lectures

Class XII - Chapter Matrices

11.5 Multiplication of a matrix by a scalar - Video Lectures

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ProfRobBob

## Sunday, May 1, 2016

### IIT JEE Mathematics Study Plan 1. Sets

R.D. Sharma, Objective Mathematics, Chapter 1

Video Lectures

1.1 Sets

1.2 Description of a set

1.3 Types of sets

1.4 Subsets

1.5 Universal set

1.6 Power set

1.7 Venn diagrams

1.8 Operations on sets

1.9 Laws of algebra of sets

1.10 More results on operations on sets

1.11 Some important results on number of elements in sets

Day 1 ( 1 May)

1.1 Sets

1.2 Description of a set

1.3 Types of sets

1.4 Subsets

1.5 Universal set

1.6 Power set

1.7 Venn diagrams

Video Lectures - Sets

Day 2 (2 May)

1.8 Operations on sets

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More Video Lectures on 1.8 Operations on Sets

1.9 Laws of algebra of sets

Day 3 (3 May)

1.10 More results on operations on sets

Day 4 (4 May)

1.11 Some important results on number of elements in sets

Day 5

Obj. Exercises 1 to 27

Day 6

Fill in the blanks 1 to 5

True/False 1 to 13

Day 7

Practice Exercises 1 to 21

For reviewing the concepts, formulas, and theorems of the chapters visit

Ch. 1. Sets - Concept Review

Updated 1 May 2016, 10 Apr 2016, 7 May 2015

Video Lectures

1.1 Sets

1.2 Description of a set

1.3 Types of sets

1.4 Subsets

1.5 Universal set

1.6 Power set

1.7 Venn diagrams

1.8 Operations on sets

1.9 Laws of algebra of sets

1.10 More results on operations on sets

1.11 Some important results on number of elements in sets

## Sets Chapter - Study Plan

(1 May to 7 May)Day 1 ( 1 May)

1.1 Sets

1.2 Description of a set

1.3 Types of sets

1.4 Subsets

1.5 Universal set

1.6 Power set

1.7 Venn diagrams

Video Lectures - Sets

Day 2 (2 May)

1.8 Operations on sets

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More Video Lectures on 1.8 Operations on Sets

1.9 Laws of algebra of sets

Day 3 (3 May)

1.10 More results on operations on sets

Day 4 (4 May)

1.11 Some important results on number of elements in sets

Day 5

Obj. Exercises 1 to 27

Day 6

Fill in the blanks 1 to 5

True/False 1 to 13

Day 7

Practice Exercises 1 to 21

For reviewing the concepts, formulas, and theorems of the chapters visit

Ch. 1. Sets - Concept Review

Updated 1 May 2016, 10 Apr 2016, 7 May 2015

### IIT JEE Mathematics Study Plan 11. Matrices

11.1 Matrix

11.2 Types of matrices

11.3 Equality of matrices

11.4 Algebra of matrices

11.5 Multiplication of a matrix by a scalar (scalar multiplication)

11.6 Subtraction of matrices (definition)

11.7 Multiplication of matrices

11.8 Transpose of a matrix

11.9 Symmetric and skew symmetric matrices

11.10 Determinants

11.11 Singular matrix

11.12 Adjoint of a matrix

11.13 Inverse of a matrix

11.14 elementary transformations of elementary operations of a matrix

11.15 Orthogonal matrix

11.16 Submatrix

11.17 Rank of a matrix

11.18 Some theorems on rank of a matrix

11.19 Equivalent matrices

11.20 Echelon form of a matrix

11.21 System of simultaneous linear equations

11.22 Solution of a non-homogeneous system of linear equations

11.23 Rank method

11.24 Solution of a homogeneous system of linear equations

Study Plan

Day 1

11.1 Matrix

11.2 Types of matrices

11.3 Equality of matrices

11.4 Algebra of matrices

Day 2

11.5 Multiplication of a matrix by a scalar (scalar multiplication)

11.6 Subtraction of matrices (definition)

11.7 Multiplication of matrices

Day 3

11.8 Transpose of a matrix

Objective Types questins 1 to 6,

Practice Exercises 1 to 10

Day 4

11.9 Symmetric and skew symmetric matrices

Ex 1 to 8

Day 5

11.10 Determinants

11.11 Singular matrix

11.12 Adjoint of a matrix

Day 6

11.13 Inverse of a matrix

11.14 elementary transformations of elementary operations of a matrix

Day 7

11.15 Orthogonal matrix

11.16 Submatrix

11.17 Rank of a matrix

11.18 Some theorems on rank of a matrix

Day 8

11.19 Equivalent matrices

11.20 Echelon form of a matrix

Objective Type Exercises 8 to 20

Day 9

11.21 System of simultaneous linear equations

11.22 Solution of a non-homogeneous system of linear equations

11.23 Rank method

Day 10

11.24 Solution of a homogeneous system of linear equations

Revision of concepts in the chapter

Day 11

OTE 21 to 40

Day 12

OTE 41 to 60

Day 13

OTE 61 to 80

Day 14

OTE 81 to 91

Fill in the blanks 1 to 17

Day 15

True/false questions 1 to 30

Day 16

Practice Exercises 11 to 20

Day 17

Practice Exercises 21 to 33

Day 18

Revision - Theory, Formulas and Difficult Problems

Day 19

Revision - Theory, Formulas and Difficult Problems

Day 20

Revision - Theory, Formulas and Difficult Problems

Updated 1 May 2016, 7 Nov 2008

11.2 Types of matrices

11.3 Equality of matrices

11.4 Algebra of matrices

11.5 Multiplication of a matrix by a scalar (scalar multiplication)

11.6 Subtraction of matrices (definition)

11.7 Multiplication of matrices

11.8 Transpose of a matrix

11.9 Symmetric and skew symmetric matrices

11.10 Determinants

11.11 Singular matrix

11.12 Adjoint of a matrix

11.13 Inverse of a matrix

11.14 elementary transformations of elementary operations of a matrix

11.15 Orthogonal matrix

11.16 Submatrix

11.17 Rank of a matrix

11.18 Some theorems on rank of a matrix

11.19 Equivalent matrices

11.20 Echelon form of a matrix

11.21 System of simultaneous linear equations

11.22 Solution of a non-homogeneous system of linear equations

11.23 Rank method

11.24 Solution of a homogeneous system of linear equations

Study Plan

Day 1

11.1 Matrix

11.2 Types of matrices

11.3 Equality of matrices

11.4 Algebra of matrices

Day 2

11.5 Multiplication of a matrix by a scalar (scalar multiplication)

11.6 Subtraction of matrices (definition)

11.7 Multiplication of matrices

Day 3

11.8 Transpose of a matrix

Objective Types questins 1 to 6,

Practice Exercises 1 to 10

Day 4

11.9 Symmetric and skew symmetric matrices

Ex 1 to 8

Day 5

11.10 Determinants

11.11 Singular matrix

11.12 Adjoint of a matrix

Day 6

11.13 Inverse of a matrix

11.14 elementary transformations of elementary operations of a matrix

Day 7

11.15 Orthogonal matrix

11.16 Submatrix

11.17 Rank of a matrix

11.18 Some theorems on rank of a matrix

Day 8

11.19 Equivalent matrices

11.20 Echelon form of a matrix

Objective Type Exercises 8 to 20

Day 9

11.21 System of simultaneous linear equations

11.22 Solution of a non-homogeneous system of linear equations

11.23 Rank method

Day 10

11.24 Solution of a homogeneous system of linear equations

Revision of concepts in the chapter

Day 11

OTE 21 to 40

Day 12

OTE 41 to 60

Day 13

OTE 61 to 80

Day 14

OTE 81 to 91

Fill in the blanks 1 to 17

Day 15

True/false questions 1 to 30

Day 16

Practice Exercises 11 to 20

Day 17

Practice Exercises 21 to 33

Day 18

Revision - Theory, Formulas and Difficult Problems

Day 19

Revision - Theory, Formulas and Difficult Problems

Day 20

Revision - Theory, Formulas and Difficult Problems

Updated 1 May 2016, 7 Nov 2008

### 1.8 Operations on Sets - Video Lectures

Union of Sets

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Intersection of Sets

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Difference of Sets and Complement of a Set

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