The Main Idea

• Definition: An augmented matrix is a way to represent a system of linear equations in matrix form.
• Structure:
  • Coefficients of variables form the main part of the matrix
  • Constants are separated by a vertical line
  • Each row represents one equation
  • Each column (before the line) represents coefficients of one variable
• Coefficient Matrix: The matrix containing only the coefficients of variables (without the constants)
• Importance of Standard Form: Equations should be in the form [latex]ax + by + cz = d[/latex] for proper alignment in the matrix
• Zero Coefficients: When a variable is missing from an equation, its coefficient is represented as [latex]0[/latex] in the matrix

The Main Idea

• Row Operations: Transformations applied to matrices that preserve the solution set of the corresponding system of equations.
• Types of Row Operations:
  • Interchanging any two rows
  • Multiplying a row by a non-zero constant
  • Adding a multiple of one row to another row
• Row-Echelon Form: A matrix form where:
  • The first non-zero element in each row (leading entry) is 1
  • Each leading 1 is to the right of the leading 1 in the row above
  • Rows with all zero elements are at the bottom
• Gaussian Elimination: A method to transform a matrix into row-echelon form using row operations.

Process of Gaussian Elimination

1. Start with the leftmost column
2. Find a non-zero entry in this column (if all zero, move to next column)
3. Move the row with this non-zero entry to the top (if not already there)
4. Make this entry a 1 by dividing the row by the entry’s value
5. Use this 1 to eliminate all other entries in this column
6. Repeat steps 1-5 for the next column, working only on rows below the current row