Matrices and Their Rank Explained

Sep 21, 2024

Engineering Mathematics - Matrices Lecture

Introduction

  • Lecturer: Dr. Gajendra Purohit
  • Subject: Engineering Mathematics
  • Topic: Matrices (Part of Linear Algebra)
  • Key topics:
    • Rank of the Matrix
    • Inconsistent and Consistent Linear Equations
    • Eigenvalues and Eigenvectors
    • Kelly-Milton Theorem

Basics of Matrices

  • Matrix Definition: Arrangement of numbers in rows and columns.
  • Representation: Matrix A as (A_{mn}) with dimensions (MxN).
    • Order: (MxN)
  • Square Matrix: Equal number of rows and columns (e.g., 2x2, 3x3).
  • Types of Matrices:
    • Row Matrix
    • Column Matrix
    • Unit Matrix

Rank of a Matrix

  • Definition: Number R is the rank of matrix (A) if:
    1. There exists at least one minor of A of order R which does not vanish.
    2. Every minor of A of order higher than R vanishes.
  • Example:
    • Matrix: ( \begin{bmatrix} 1 & 2 & 3 \ 2 & 4 & 8 \ 2 & 4 & 6 \end{bmatrix} )
    • Sub-matrices (minors) have non-zero determinants while the full matrix determinant is zero.
  • Trick to Determine Rank:
    • If two lines in a matrix are similar, count as one.
    • Use row transformations to simplify and find rank.

Methods to Find Rank

  1. Determinant Method (Minor Method)
    • Not always usable for non-square matrices.
  2. Row Transformation
    • Used when determinant method is infeasible.
    • Steps:
      • Use top-left element to make below elements zero.
      • Transform row to create zeros.
  3. Short Trick:
    • For 3x3 or 3x4 matrices, make two zeros using a guiding element, then another zero.

Examples and Steps

  • Example 1: 3x4 Matrix Rank
  • Example 2: 4x4 Matrix Rank
    • Utilize column transformations for normal form.
    • Objective: Reduce to unit matrix with zeros elsewhere.

Normal Form

  • Definition: Matrix in reduced form with unit matrix and zeros.
  • Process: Reduce matrix to normal form to confirm rank.

Conclusion

  • Rank is the number of non-zero linearly independent rows.
  • Consistent and inconsistent linear equations depend on matrix rank.
  • Upcoming topics: Consistency of linear equations.

This lecture provided an overview of matrices focusing on the rank of the matrix, methods to determine it, and transforming matrices to their normal form. Upcoming lectures will explore linear equations and their consistency. Stay tuned for more insights into engineering mathematics.