Topics
Mathematical Logic
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality 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
Trigonometric Functions
Pair of Straight Lines
Vectors
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivative of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- 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 Two Curves
- Overview of Application of Definite Integration
Differential Equations
Probability Distributions
- 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
- 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.
Video link : https://youtu.be/lBnSOCO-9EI
