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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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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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^-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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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^-1A^-1

6.If A,B,C are invertible matrices then
(ABC) ^-1 = C^-1B^-1A^-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-2A

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

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

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^TA^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

11.7 Multiplication of matrices - Video Lectures
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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

1.9 Laws of algebra of sets - Video Lectures

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eduarrow.com

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

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

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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