Topics
Relations and Functions
Relations and Functions
Algebra
Inverse Trigonometric Functions
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
Determinants
Vectors and Three-dimensional Geometry
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
Sets
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
- Properties of Definite Integrals
- Overview of Integrals
Applications of the Integrals
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- Solution of a Differential Equation
- Forms of Solving Differential Equations> Variable Separable
- Forms of Solving Differential Equations> Homogeneous Differential Equations
- Forms 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
- Algebra of Vectors > Scalar Multiplication
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Vector Algebra
- Product of Two Vectors > Scalar (Dot) Product
- Product of Two Vectors > Vector (Cross) Product
- Overview of Vectors
Three - Dimensional Geometry
Linear Programming
Probability
Definition: Equality of Matrices
Two matrices are equal if and only if:
- They have the same order (same number of rows and columns), and
- Their corresponding elements are equal.
i.e
If \[ A = [a_{ij}]_{m \times n} \quad \text{and} \quad B = [b_{ij}]_{m \times n}, \]
then \[ A = B \iff a_{ij} = b_{ij} \text{ for all } i, j, \]
Example:
\[A=
\begin{bmatrix}
2 & & 3 \\
1 & & 5
\end{bmatrix}\mathrm{and} B=
\begin{bmatrix}
2 & & 3 \\
1 & & 5
\end{bmatrix}\]
Example: 1
Find the values of a, b, c, and d from the following equation:
\[\begin{bmatrix} 2a+b & a-2b \\ 5c-d & 4c+3d \end{bmatrix}= \begin{bmatrix} 4 & -3 \\ 11 & 24 \end{bmatrix}\]
Solution: By the equality of two matrices, equating the corresponding elements, we get
2a + b = 4 5c – d = 11
a – 2b = – 3 4c + 3d = 24
Solving these equations, we get
a = 1, b = 2, c = 3 and d = 4
Example 2
\[ P = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \qquad Q = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix} \]
Both matrices have order \[2 \times 2\], but the element in position \[(2,2)\] is different: \[4 \neq 5\]. Therefore, \[P \neq Q\].
Remember
Matrices containing the same numbers are not necessarily equal if the numbers occur in different positions.
\[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \neq \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \]
Key Points: Equality of Matrices
-
Equality of matrices is possible only when the order is the same.
-
Corresponding elements must be compared position by position.
-
If even one corresponding entry differs, the matrices are not equal.
