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
- Derivatives of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivative of Implicit Functions
- Derivatives of Functions in Parametric Forms
- 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
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Applications of Differential Equation
- Solution of a Differential Equation
- Overview of 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
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
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Matrix multiplication is row-by-column, not term-wise.
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Product AB exists only if columns of A = rows of B.
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If A is \[m \times n\] and B is \[n \times p\], then AB is \[m \times p\].
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In general, \[AB \neq BA\], and sometimes one product may not even be defined.
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Matrix multiplication is associative and distributive over addition.
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Identity matrix acts as a multiplicative identity: AI = IA = A.
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Zero matrix absorbs multiplication: AO = OA = O.
