Linear Algebra BAS113 Lecture 3 Quiz

Linear Algebra - BAS113

• Course instructors: Dr. Amira A. Allam and Dr. Mahmoud Owais
• Course title: Linear Algebra
• Course code: BAS113
• University: Sphinx University

Lecture 3: Matrix Operation 2

• Learning Objectives: Matrices, Elementary Row Operations, Inverse matrices

Elementary Row Operations

• Definition: Elementary row operations are methods to transform matrices. These include:
  • Interchanging two rows
  • Multiplying a row by a non-zero constant
  • Replacing a row by the sum of itself and a multiple of another row.
• Row Equivalent Matrices: Matrices A and B are said to be row equivalent (A ~ B) if one can be obtained from the other using a finite sequence of elementary row operations.

Row-Echelon Form

• Definition: A matrix is in row-echelon form if it satisfies certain properties:

  • The first entry of the first column is non-zero (leading entry is 1 if possible) and all other entries in that column are zeros.
  • The first column with a non-zero entry that is not in the first has its non-zero entry in the second row; every entry below that entry in that column is also zero.
  • This pattern continues for subsequent columns. The first non-zero in each column is to the right of the previous first non-zero entry in the previous column.

• Example Matrices in Row-Echelon Form:

  • [1 4 5] [1 3 5] [0 1 0] [0 1 0] [0 0 1] [0 0 1]

• Example Matrix not in Row Echelon Form: [4 0 1] [0 0 0] [0 0 0]

Reduced Row Echelon Form

• Definition: A matrix is in reduced row-echelon form if it additionally satisfies the following conditions.

  • If a row is not all zeros, the leading entry is 1
  • Any column containing the leading entry of some row contains only zeros except for the leading entry, which is 1.

• Example Matrices in Reduced Row-Echelon Form:

  • [1 0 2] [1 4 7 0] [0 1 0] [0 1 1 0] [0 0 1] [0 0 1 0]
  • [1 0 0] [0 1 0] [0 0 1]

• Example Matrices not in Reduced Row Echelon Form: [1 1 0] [0 0 0]

  [1 0 1]
  [0 1 1]
  

Inverse of a Matrix

• Definition: The inverse of a square matrix A, denoted as A⁻¹, is another square matrix with the same dimensions that satisfies the following: A x A⁻¹ = I (identity matrix) and A⁻¹ x A = I.
• Example:
• A matrix is invertible if there exists an inverse.
• How to find the inverse: Use elementary Row Operations to transform the given matrix and the identity to a row-echelon form.

Test your knowledge on matrix operations and elementary row operations in this quiz based on Lecture 3 of the Linear Algebra course. Understand concepts such as row equivalent matrices and row-echelon form. Perfect for students of Sphinx University.