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
Definition
Defination: The adjoint of a square matrix A = `[a_(ij)]_(n × n)` is defined as the transpose of the matrix `[A_(ij)]_(n × n)`, where `A_(ij)` is the cofactor of the element `a_(ij)`. Adjoint of the matrix A is denoted by adj A.
Let A = `[(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)]`
Then adj A = Transpose of `[(A_11,A_12,A_13),(A_21,A_22,A_23),(A_31,A_32,A_33)] = [(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]`
Theorem
If A be any given square matrix of order n, then
A(adj A) = (adj A) A = |A| I ,
where I is the identity matrix of order n
Verification
Let A =`[(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)]` , then adj A =` [(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]`
Since sum of product of elements of a row (or a column) with corresponding cofactors is equal to |A| and otherwise zero, we have
A (adj A) = `|(|A|,0,0),(0,|A|,0),(0,0,|A|)| = |A| [(1,0,0),(0,1,0),(0,0,1)]`
Similarly, we can show (adj A) A = A I
Hence A (adj A) = (adj A) A = A I
Notes
Definition : A square matrix A is said to be singular if A = 0.
For example, the determinant of matrix A = `[(1,2),(4,8)]` is zero
Hence A is a singular matrix.
Definition : A square matrix A is said to be non-singular if A ≠ 0
Let A = `[(1,4),(3,2)]`. Then |A| = `|(1,4)(3,2)|` = 4 - 6 = 2 ≠ 0
Hence A is a nonsingular matrix.
We state the following theorems without proof.
Theorem : If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.
Theorem: The determinant of the product of matrices is equal to product of their respective determinants, that is, |AB| = |A| |B| , where A and B are square matrices of the same order
Remark: We know that (adj A) A = |A| I = `[(|A|,0,0),(0,|A|,0),(0,0,|A|)]` , |A| ≠ 0
Writing determinants of matrices on both sides, we have
`|(adj A)A| = |(|A|,0,0),(0,|A|,0),(0,0,|A|)|`
i.e,. `|(adjA)| |A| = |A|^3 |(1,0,0),(0,1,0),(0,0,1)|`
i.e. `|(adj A)| |A| = |A|3 (1) i.e. |(adj A)| = |A|2`
i.e. `|(adj A)| = |A|^2`
In general, if A is a square matrix of order n, then |adj(A)| = |A|^(n – 1).
Theorem: A square matrix A is invertible if and only if A is nonsingular matrix.
Proof: Let A be invertible matrix of order n and I be the identity matrix of order n. Then, there exists a square matrix B of order n such that
AB = BA = I
Now AB = I. So |AB| = I or |A| |B| = 1 (since |I| =1, |AB|=|A||B|)
This gives |A| ≠ 0.
Hence A is nonsingular.
Conversely, let A be nonsingular. Then A ≠ 0
Now A (adj A) = (adj A) A = |A| I (Theorem 1)
or `A(1/|A| adj A) = (1/|A| adj A) A = I`
or `AB = BA = I` ,where `B = 1/|A| adj A`
Thus, A is invertible and `A^-1 = 1/ |A|adj A`
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Related QuestionsVIEW ALL [169]
Read the following passage:
|
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹190. Also, Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹250. |
Based on the above information, answer the following questions:
- Convert the given above situation into a matrix equation of the form AX = B. (1)
- Find | A |. (1)
- Find A–1. (2)
OR
Determine P = A2 – 5A. (2)
A furniture factory uses three types of wood namely, teakwood, rosewood and satinwood for manufacturing three types of furniture, that are, table, chair and cot.
The wood requirements (in tonnes) for each type of furniture are given below:
| Table | Chair | Cot | |
| Teakwood | 2 | 3 | 4 |
| Rosewood | 1 | 1 | 2 |
| Satinwood | 3 | 2 | 1 |
It is found that 29 tonnes of teakwood, 13 tonnes of rosewood and 16 tonnes of satinwood are available to make all three types of furniture.
Using the above information, answer the following questions:
- Express the data given in the table above in the form of a set of simultaneous equations.
- Solve the set of simultaneous equations formed in subpart (i) by matrix method.
- Hence, find the number of table(s), chair(s) and cot(s) produced.
