Topics
Mathematical Logic
- Concept of Statements
- Truth Value of Statement
- Logical Connective, Simple and Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Duality
- Quantifier and Quantified Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
Matrices
- Elementry Transformations
- Properties of Matrix Multiplication
- Application of Matrices
- Applications of Determinants and Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle
- Inverse Trigonometric Functions
- Overview of Trigonometric Functions
Pair of Straight Lines
- Combined Equation of a Pair Lines
- Homogeneous Equation of Degree Two
- Angle between lines represented by ax2 + 2hxy + by2 = 0
- General Second Degree Equation in x and y
- Equation of a Line in Space
- Overview of Pair of Straight Lines
Vectors
Line and Plane
- Vector and Cartesian Equations of a Line
- Distance of a Point from a Line
- Distance Between Skew Lines and Parallel Lines
- Equation of a Plane
- Angle Between Planes
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Overview of Line and Plane
Linear Programming
Differentiation
- Differentiation
- Derivatives of Composite Functions - Chain Rule
- Geometrical Meaning of Derivative
- Derivatives of Inverse Functions
- Logarithmic Differentiation
- Derivatives of Implicit Functions
- Derivatives of Parametric Functions
- Higher Order Derivatives
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by the Curve, Axis and Line
- Area Between Two Curves
- Overview of Application of Definite Integration
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Homogeneous Differential Equations
- Linear Differential Equations
- Application of Differential Equations
- Solution of a Differential Equation
- Overview of Differential Equations
Probability Distributions
- Random Variables and Its Probability Distributions
- Types of Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
Binomial Distribution
- Bernoulli Trial
- Binomial Distribution
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Bernoulli Trials and Binomial Distribution
- Overview of Binomial Distribution
- Consistent System
- Inconsistent System
- Solution of a system of linear equations using the inverse of a matrix
Notes
Consistent system: A system of equations is said to be consistent if its solution (one or more) exists.
Inconsistent system: A system of equations is said to be inconsistent if its solution does not exist.
Solution of system of linear equations using inverse of a matrix:
Consider the system of equations
`a_1 x + b_1 y + c_1 z = d_1`
`a_2 x + b_2 y + c_2 z = d_2`
`a _3 x + b_3 y + c_3 z = d_3`
Let A =` [(a_1,b_1,c_1),(a_1,b_2,c_2),(a_3,b_3,c_3)] , X = [x,y,z]` and
B =` [(d_1),(d_2),(d_3)]`
Then, the system of equations can be written as, AX = B, i.e.,
`[(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)] [(x),(y),(z)] = [(d_1),(d_2),(d_3)]`
Case I: If A is a nonsingular matrix, then its inverse exists. Now
AX = B
or `A^(–1)` (AX) = `A^(–1)` B (premultiplying by A–1)
or `(A^(–1)A)` X = `A^(–1)` B (by associative property)
or I X = `A^(–1)` B
or X = `A^(–1)` B
This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as Matrix Method.
Case II: If A is a singular matrix, then |A| = 0.
In this case, we calculate (adj A) B.
If (adj A) B ≠ O, (O being zero matrix), then solution does not exist and the system of equations is called inconsistent.
If (adj A) B = O, then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution.
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