Rank of Matrices

Start of Lecture

• Welcomed all students.
• There was a slight delay due to technical issues.

Introduction to the Topic

• Today's topic: Rank of Matrices.
• Understanding rank will help in comprehending future topics.

Elementary Transformation

• Knowledge of elementary transformations is essential for calculating rank.
• Three types of elementary transformations:
  • Row Swap: Swapping one row with another. (R1 ↔ R2)
  • Scalar Multiplication: Multiplying a row by a scalar. (R1 = k * R1)
  • Row Addition: Adding one row to another. (R1 = R1 + k * R2)

Definition of Rank

• Rank is an intrinsic property of a matrix that does not change.
• If a matrix B is obtained from A via elementary transformations, then A and B will have the same rank.

Calculating Rank

• To calculate rank, first bring the matrix to row-echelon form.
• In row-echelon form, zero rows are placed at the bottom, and non-zero rows are arranged sequentially.
• The rank is determined by the number of non-zero rows.

Important Points on Rank

• The rank of a row matrix and column matrix can be either 0 or 1.
• When all elements are zero, the rank is 0.
• If at least one element is non-zero, the rank is 1.

Procedure

• To determine the rank of a matrix:
  1. First, identify zero rows and place them at the bottom.
  2. Count the number of non-zero rows.
  3. This number will be the rank value.

Question and Answer Session

• Some questions and answers were conducted in the lecture to ensure student clarity.

Closing

• Next session time: Tomorrow at 4:00 PM.
• Thanked all students.