Matrix Algebra and Rank

This module covers both algebraic and graphical approaches to evaluating the characteristics of matrices. A detailed account on row/column reductions to Echelon form and hence Rank of a matrix and its graphical understanding. LU decomposition is discussed as a special case of matrix-matrix multiplication. Significant number of model questions paper problems have been solved to give a comprehensive understanding of the examiner’s perspective.

Salient features of this module are:

• Pre-requisites to Fundamental Matrix Theory
• Graphical visualization
• Concept testing
• Unravelling examiners mind
• Model question paper

Course Contents: 

Pre-requisites to Matrix Algebra and Rank

• Types of Matrices
  • Types of Matrices – Part 1
  • Types of Matrices – Part 2
  • Types of Matrices – Part 3
• Understanding Determinant
  • Determinant of a 3×3 matrix
  • Determinant of a 4×4 matrix
• Vector Algebra for Matrix Multiplication
  • Dot and Cross Product – Part 1
  • Dot and Cross Product – Part 2
• Graphical Perspection of a Determinant
  • Perspective – 1
  • Perspective – 2
• Inverse of a Matrixe
  • Introduction – 2X2 – Matrix
  • Example – 2X2 – Matrix
  • General procedure and Incidental facts – 3X3 Matrix
  • Observations
  • Applications
• Matrix-Vector Multiplication
  • Introduction – Perspective 1 – Linear Algebra
  • Introduction – Perspective 2 – Linear Algebra
  • Quiz 1
  • Quiz 2
  • Quiz 3
• Matrix-Matrix Multiplication
  • Introduction – Perspective – Linear Algebra
  • Introduction – Alternate Method
  • Quiz 1
  • Quiz 2
  • Quiz 3
  • Quiz 4
  • Quiz 5

Matrix algebra & Rank

• Rank of a Matrix
  • Concept of Rank – (Basis definition of Rank)
  • Concept of Rank – (Linearly dependent and independent vectors)
  • Concept of Rank – (Graphical Understanding – 1)
  • Linear Algebra – (Rank – Graphical Understanding – 2)
  • Linear Algebra – (Rank – Linear Algebra/Transformation Perspective)
  • Linear Algebra – (Final observations/conclusions)
  • Rank/Row and Column Reduction – Problems: (problem 1 – Row reduction)
  • Rank/Row and Column Reduction – Problems: (problem 2 – Column reduction)
  • Rank/Row and Column Reduction – (Understanding the Echelon and observations)
  • Rank/Row and Column Reduction – (Understanding the Echelon and Normal form of a matrix)
  • Rank/Echelon form – Problems: (Problem 1)
  • Rank/Echelon form – Problems: (Problem 2)
  • Rank/Echelon form – Problems: (Problem 3)
  • Rank/Echelon form – Problems: (Problem 4)
  • Rank/Echelon form – Problems: (Problem 5)
  • Rank/Echelon form – Problems: (Problem 6)
  • Rank/Echelon form – Problems: (Problem 7)
  • Rank/Echelon form – Problems: (Problem 8)