Engineering Mathematics I - Lecture Notes
Instructor: Dr. Gajendra Purohit
Introduction
- Course: Engineering Mathematics I
- Initial Topic: Matrices
- Part of Linear Algebra
- Key Topics in Matrices:
- Rank of a Matrix
- Inconsistent and Consistent Linear Equations
- Eigenvalues and Eigenvectors
- Cayley Hamilton Theorem
Matrices Basics
- Definition: A matrix is represented in form 'm x n', where:
- m = number of rows
- n = number of columns
- Square Matrix: A matrix with equal numbers of rows and columns (e.g., 2x2, 3x3)
- Types of Matrices: Row matrix, Column matrix, Unit matrix
Rank of a Matrix
Basic Definition
- Rank is defined by the number of different rows in a matrix.
- Examples:
- 3x3 matrix:
- All different rows: Rank = 3
- Two same rows: Rank = 2
- All same rows: Rank = 1
Sub-matrices and Minors
- Matrices can have sub-matrices called 'Minors'
- A minor's determinant must be non-zero for determining rank
Methods to Find Rank
-
Determinant Method (Minor Method)
- Find rank using sub-matrices
- Limited to square matrices
-
Row and Column Transformation
- More adaptable method
- Steps involve transforming the matrix into Echelon form
- Example: If last line is non-zero, rank is the number of non-zero rows
-
Normal Form Reduction
- Used when rank is required in normal form
- Transform matrix into a unit matrix of the same rank (e.g., rank 2 results in a 2x2 unit matrix)
Practical Examples
- Several examples were shown:
- 3x4 Matrix Example
- 4x4 Matrix Example
- Key learning involves transformation to find a matrix’s rank
Conclusion
- Today's focus: Rank of a Matrix
- Upcoming Topic: Consistent and Inconsistent Linear Equations
- Dependent on understanding of rank
- Summary:
- Rank depends on the uniqueness of rows in a matrix
- Transformation methods and tricks explained
Future Sessions
- More videos on Engineering Mathematics to follow
Thank you for attending the lecture. Stay tuned for more insights into Engineering Mathematics.