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
- 1st, 2nd and 3rd Row
- 1st, 2nd and 3rd Columns
- Expansion along the first Row (R1)
- Expansion along the second row (R2)
- Expansion along the first Column (C1)
Notes
Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows `(R_1, R_2 and R_3)` and three columns `(C_1, C_2 and C_3)` giving the same value as shown below. Consider the determinant of square matrix A = `[a_(ij)]_(3 × 3)`
i.e. |A| = `|(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|`
Expansion along first Row `(R_1)`
Step 1: Multiply first element `a_11` of `R_1` by `(–1)^((1 + 1)) [(–1)^("sum of suffixes in" a_11)]` and with the second order determinant obtained by deleting the elements of first row `(R_1)` and first column `(C_1)` of
| A | as `a_11` lies in `R_1` and `C_1`,
i.e., `(-1)^(1+1) a_11|(a_22,a_23),(a_32,a_33)|`
Step 2: Multiply 2nd element `a_12` of `R_1` by` (–1)^(1 + 2) [(–1)^("sum of suffixes in" a_12)]` and the second order determinant obtained by deleting elements of first row `(R_1)` and 2nd column `(C_2)` of | A | as
`a_12` lies in` R_1` and` C_2,`
i.e., `(-1)^(1+2) a_12|(a_21,a_23),(a_31,a_33)|`
Step 3: Multiply third element `a_13` of `R_1` by `(–1)^(1 + 3) [(–1)^("sum of suffixes in" a_13)]` and the second order determinant obtained by deleting elements of first row `(R_1)` and third column`(C_3)` of | A | as `a_13` lies in` R_1` and `C_3`,
i.e., `(-1)^(1+3) a_13 |(a_21,a_22),(a_31,a_33)|`
Step 4: Now the expansion of determinant of A, that is, | A | written as sum of all three terms obtained in steps 1, 2 and 3 above is given by
det A = |A| = `(–1)^(1 + 1) a_11 |(a_22,a_23),(a_32,a_33)| + (-1)^(1+2) a_12 |(a_21,a_23),(a_31,a_33)| + (-1)^(1+3) a_13 |(a_21,a_22)(a_31,a_32)|`
|A|
= `a_11 (a_22 a_33 – a_32 a_23) – a_12 (a_21 a_33 – a_31 a_23)
+ a_13 (a_21 a_32 – a_31 a_22)`
= `a_11 a_22 a_33 – a_11 a_32 a_23 – a_12 a_21 a_33 + a_12 a_31 a_23 + a_13 a_21 a_32 – a_13 a_31 a_22 ` ... (1)
Expansion along second row `(R_2)`:
|A| = `|(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|`
Expanding along R2, we get
|A| = `(-1)^(2+1) a_21 |(a_12,a_13),(a_32,a_33)| +(-1)^(2+2) a_22 |(a_11,a_13),(a_31,a_33)| + (-1)^(2+3) a_23 |(a_11,a_12),(a_31,a_32)|`
=` – a_21 (a_12 a_33 – a_32 a_13) + a_22 (a_11 a_33 – a_31 a_13) – a_23 (a_11 a_32 – a_31 a_12)`
| A | = `– a_21 a_12 a_33 + a_21 a_32 a_13 + a_22 a_11 a_33 – a_22 a_31 a_13 – a_23 a_11 a_32 + a_23 a_31 a_12 `
= `a_11 a_22 a_33 – a_11 a_23 a_32 – a_12 a_21 a_33 + a_12 a_23 a_31 + a_13 a_21 a_32 – a_13 a_31 a-22 ` ... (2)
Expansion along first Column `(C_1)`:
|A| = `|(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|`
By expanding along C1, we get
|A| =` a_11 (-1)^(1+1) |(a_22,a_23),(a_32,a_33)| + a_21 (-1)^(2+1) |(a_12,a_13)(a_32,a_33)| + a_31(-1)^(3+1) |(a_12,a_13)(a_22,a_23)|`
=` a_11 (a_22 a_33 – a_23 a_32) – a_21 (a_12 a_33 – a_13 a_32) + a_31 (a_12 a_23 – a_13 a_22)`
| A | = `a_11 a_22 a_33 – a_11 a_23 a_32 – a_21 a_12 a_33 + a_21 a_13 a_32 + a_31 a_12 a_23 – a_31 a_13 a_22`
= `a_11 a_22 a_33 – a_11 a_23 a_32 – a_12 a_21 a_33 + a_12 a_23 a_31 + a_13 a_21 a_32 – a_13 a_31 a_22` ... (3)
Clearly, values of |A| in (1), (2) and (3) are equal. It is left as an exercise to the reader to verify that the values of |A| by expanding along `R_3`, `C_2` and `C_3` are equal to the value of |A| obtained in (1), (2) or (3).
Hence, expanding a determinant along any row or column gives same value.
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