Topics
Relations and Functions
Relations and Functions
Inverse Trigonometric Functions
- Basics of Inverse Trigonometric Functions
- Domain, Range & Principal Value
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions > Self-adjusting Property
- Overview of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions > Reciprocal Property
- Properties of Inverse Trigonometric Functions > Negative Argument Property
- Properties of Inverse Trigonometric Functions > Complementary Property
- Properties of Inverse Trigonometric Functions > Conversion Property
- Properties of Inverse Trigonometric Functions > Addition & Subtraction Formula for Inverse Tangent
- Properties of Inverse Trigonometric Functions > Double-angle Property
- Properties of Inverse Trigonometric Functions > Triple-angle Property
- Properties of Inverse Trigonometric Functions > Addition–Subtraction Formula for Inverse Sine & Cosine
Algebra
Calculus
Matrices
- Concept of Matrices
- Types of Matrices
- Equality of Matrices
- Operations on Matrices> Addition and Subtraction of Matrices
- Operations on Matrices>Scalar Multiplication
- Operations on Matrices> Matrix Multiplication
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Invertible Matrices
- Overview of Matrices
Vectors and Three-dimensional Geometry
Determinants
Continuity and Differentiability
- Continuous and Discontinuous Functions
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions
- Differentiation of Implicit Functions
- Derivative of Inverse Trigonometric Function
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Overview of Continuity and Differentiability
Linear Programming
Applications of Derivatives
Probability
Integrals
- Integration
- Integration as an Inverse Process of Differentiation
- Properties of Indefinite Integral
- Methods of Integration> Integration by Substitution
- Methods of Integration>Integration Using Trigonometric Identities
- Methods of Integration> Integration Using Partial Fraction
- Methods of Integration> Integration by Parts
- Integrals of Some Particular Functions
- Definite Integrals
- Fundamental Theorem of Integral Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Overview of Integrals
Sets
Applications of the Integrals
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Methods of Solving Differential Equations> Variable Separable Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Overview of Differential Equations
Vectors
- Basic Concepts of Vector Algebra
- Direction Ratios, Direction Cosine & Direction Angles in Vector
- Types of Vectors in Algebra
- Algebra of Vectors > Addition & Subtraction of Two Vectors
- Multiplication in Vector Algebra
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Vector Algebra
- Product of Two Vectors > Scalar (Dot) Product
- Overview of Vectors
Three - Dimensional Geometry
Linear Programming
Probability
Introduction
Matrix multiplication is a fundamental operation in algebra that helps us represent and solve systems of equations, perform coordinate transformations, and model real-life situations like network flows and data transformations.
Definition: Matrix Multiplication
Let \[A = [a_{ij}]\] be an \[m \times n\] matrix and \[B = [b_{jk}]\] be an \[n \times p\] matrix.
Then the product C = AB is an \[m \times p\] matrix \[C = [c_{ik}]\], where each entry \[c_{ik}\] is given by:
Properties
| Property | Rule / Formula |
|---|---|
| Non-commutativity | In general, \[AB \neq BA\] |
| Associativity | \[(AB)C = A(BC)\] |
| Left distributive law | \[A(B + C) = AB + AC\] |
| Right distributive law | \[(A + B)C = AC + BC\] |
| Multiplication by zero matrix | AO = O and OA = O |
| Identity matrix property | AI = IA = A |
| Cancellation law | From AB = AC, we cannot always conclude B = C |
Example 1
If \[\mathrm{A=} \begin{bmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0 \end{bmatrix},\mathrm{B=} \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix},\mathrm{C=} \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix}\]
Calculate AC, BC and (A + B)C. Also, verify that (A + B)C = AC + BC
Solution: Now, \[\mathbf{A}+\mathbf{B}= \begin{bmatrix} 0 & 7 & 8 \\ -5 & 0 & 10 \\ 8 & -6 & 0 \end{bmatrix}\]
So \[\mathrm{(A+B)C}= \begin{bmatrix} 0 & 7 & 8 \\ -5 & 0 & 10 \\ 8 & -6 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix}= \begin{bmatrix} 0-14+24 \\ -10+0+30 \\ 16+12+0 \end{bmatrix}= \begin{bmatrix} 10 \\ 20 \\ 28 \end{bmatrix}\]
Further \[\mathrm{AC}= \begin{bmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix}= \begin{bmatrix} 0-12+21 \\ -12+0+24 \\ 14+16+0 \end{bmatrix}= \begin{bmatrix} 9 \\ 12 \\ 30 \end{bmatrix}\]
and \[\mathrm{BC}= \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix}= \begin{bmatrix} 0-2+3 \\ 2+0+6 \\ 2-4+0 \end{bmatrix}= \begin{bmatrix} 1 \\ 8 \\ -2 \end{bmatrix}\]
So \[\mathrm{AC}+\mathrm{BC}= \begin{bmatrix} 9 \\ 12 \\ 30 \end{bmatrix}+ \begin{bmatrix} 1 \\ 8 \\ -2 \end{bmatrix}= \begin{bmatrix} 10 \\ 20 \\ 28 \end{bmatrix}\]
Clearly, (A + B) C = AC + BC
CISCE: Class 10
Key Points: Matrix Multiplication
-
Matrix multiplication is row-by-column, not term-wise.
-
Product AB exists only if columns of A = rows of B.
-
If A is \[m \times n\] and B is \[n \times p\], then AB is \[m \times p\].
-
In general, \[AB \neq BA\], and sometimes one product may not even be defined.
-
Matrix multiplication is associative and distributive over addition.
-
Identity matrix acts as a multiplicative identity: AI = IA = A.
-
Zero matrix absorbs multiplication: AO = OA = O.
