Topics
Relations and Functions
Relations and Functions
Inverse Trigonometric Functions
Algebra
Calculus
Matrices
- Introduction of Matrices
- Order of a Matrix
- Types of Matrices
- Equality of Matrices
- Introduction of Operations on Matrices
- Addition of Matrices
- Multiplication of a Matrix by a Scalar
- Properties of Matrix Addition
- Properties of Scalar Multiplication of a Matrix
- Multiplication of Matrices
- Properties of Multiplication of Matrices
- Transpose of a Matrix
- Properties of Transpose of the Matrices
- Symmetric and Skew Symmetric Matrices
- Invertible Matrices
- Inverse of a Matrix by Elementary Transformation
- Multiplication of Two Matrices
- Negative of Matrix
- Subtraction of Matrices
- Proof of the Uniqueness of Inverse
- Elementary Transformations
- Matrices Notation
Determinants
- Introduction of Determinant
- Determinants of Matrix of Order One and Two
- Determinant of a Matrix of Order 3 × 3
- Area of a Triangle
- Minors and Co-factors
- Inverse of a Square Matrix by the Adjoint Method
- Applications of Determinants and Matrices
- Elementary Transformations
- Properties of Determinants
- Determinant of a Square Matrix
- Rule A=KB
Vectors and Three-dimensional Geometry
Linear Programming
Continuity and Differentiability
- Concept of Continuity
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions - Chain Rule
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Derivative - Exponential and Log
- Proof Derivative X^n Sin Cos Tan
- Infinite Series
- Higher Order Derivative
- Continuous Function of Point
- Mean Value Theorem
Applications of Derivatives
- Introduction to Applications of Derivatives
- Rate of Change of Bodies or Quantities
- Increasing and Decreasing Functions
- Maxima and Minima
- Maximum and Minimum Values of a Function in a Closed Interval
- Simple Problems on Applications of Derivatives
- Graph of Maxima and Minima
- Approximations
- Tangents and Normals
Probability
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Some Properties of Indefinite Integral
- Methods of Integration: Integration by Substitution
- Integration Using Trigonometric Identities
- Integrals of Some Particular Functions
- Methods of Integration: Integration Using Partial Fractions
- Methods of Integration: Integration by Parts
- Fundamental Theorem of Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Definite Integrals
- Indefinite Integral Problems
- Comparison Between Differentiation and Integration
- Geometrical Interpretation of Indefinite Integrals
- Indefinite Integral by Inspection
- Definite Integral as the Limit of a Sum
- Evaluation of Simple Integrals of the Following Types and Problems
Sets
- Sets
Applications of the Integrals
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Linear Differential Equations
- Homogeneous Differential Equations
- Solutions of Linear Differential Equation
- Differential Equations with Variables Separable Method
- Formation of a Differential Equation Whose General Solution is Given
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
Vectors
- Introduction of Vector
- Basic Concepts of Vector Algebra
- Direction Cosines
- Vectors and Their Types
- Addition of Vectors
- Properties of Vector Addition
- Multiplication of a Vector by a Scalar
- Components of Vector
- Vector Joining Two Points
- Section Formula
- Vector (Or Cross) Product of Two Vectors
- Scalar (Or Dot) Product of Two Vectors
- Projection of a Vector on a Line
- Geometrical Interpretation of Scalar
- Scalar Triple Product of Vectors
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Magnitude and Direction of a Vector
- Vectors Examples and Solutions
- Introduction of Product of Two Vectors
Three - Dimensional Geometry
- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Relation Between Direction Ratio and Direction Cosines
- Equation of a Line in Space
- Angle Between Two Lines
- Shortest Distance Between Two Lines
- Three - Dimensional Geometry Examples and Solutions
- Equation of a Plane Passing Through Three Non Collinear Points
- Intercept Form of the Equation of a Plane
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Angle Between Two Planes
- Vector and Cartesian Equation of a Plane
- Equation of a Plane in Normal Form
- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
- Distance of a Point from a Plane
- Plane Passing Through the Intersection of Two Given Planes
Linear Programming
Probability
- Introduction of Probability
- Conditional Probability
- Properties of Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ Theorem
- Variance of a Random Variable
- Probability Examples and Solutions
- Random Variables and Its Probability Distributions
- Mean of a Random Variable
- Bernoulli Trials and Binomial Distribution
- Non-commutativity of multiplication of matrices
- Zero matrix as the product of two non zero matrices
Notes
The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. Let A = `[a_(ij)]` be an m × n matrix and B = `[b_(jk)]` be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p. To get the `(i, k)th`element cik of the matrix C, we take the ith row of A and kth column of B, multiply them elementwise and take the sum of all these products. In other words, if A = `[a_(ij)]_(m × n)`, B = `[b_(jk)]_(n × p)`, then the ith row of A is `[a_(i1) a_(i2) ... a_("in")]` and the `k^(th)` column of
B is `[(b_(1k)) , (b_(2k)), (b_(nk))]`
, then `c_(ik) = a_(i1) b_(1k) + a_(i2) b_(2k) + a_(i3) b_(3k) + ... + a_("in") b_(nk)`
=\[\displaystyle\sum_{j=1}^{n} a_{ij} b_{jk}\].
Non-commutativity of multiplication of matrices:
The below example that even if AB and BA are both defined, it is not necessary that AB = BA.
If A = `[(1,0),(0,-1)]` and B =`[(0,1),(1,0)]` ,
then AB `[(0,1),(-1,0)]`
and BA = `[(0,-1),(1,0)]`.
Clearly AB ≠ BA.
Thus matrix multiplication is not commutative.
Zero matrix as the product of two non zero matrices:
The real numbers a, b if ab = 0, then either a = 0 or b = 0. This need not be true for matrices, we will observe this through an example.
find AB ,if A = `[(0,-1),(0,2)]` and B = `[(3,5),(0,0)]`
We have AB = `[(0,-1),(0,2)][(3,5),(0,0)]`
=`[(0,0),(0,0)]`
Thus, if the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.
