11.6 Solving Systems with Gaussian Elimination

Gaussian elimination is a powerful method for solving systems of linear equations. It transforms equations into augmented matrices, then uses row operations to simplify them into row echelon form. This process reveals solutions and helps classify systems as having unique, infinite, or no solutions.

Understanding Gaussian elimination is crucial for tackling complex linear systems. It provides a systematic approach to solving equations, making it an essential tool in algebra, engineering, and data analysis. Mastering this technique opens doors to more advanced mathematical concepts.

Gaussian Elimination

Augmented matrices from equations

• Represents a system of linear equations in matrix form
• Each row corresponds to an equation
• Entries in each row are coefficients (numbers multiplying variables) of variables in the equation
• Last column contains constants from right-hand side of equations
• Creating an augmented matrix from a system of equations:
  • Write coefficients of each variable in same order for each equation
  • Add constants from right-hand side as last column
  • Separate coefficients and constants with a vertical line
• Creating a system of equations from an augmented matrix:
  • Each row represents an equation
  • Entries in each row (except last column) are coefficients of variables
  • Last column contains constants from right-hand side of equations

Row operations for echelon form

• Row echelon form is a matrix format where:
  • First non-zero entry in each row (pivot) is 1
  • Pivot of each row is located to the right of pivot in row above
  • All other entries in column below a pivot are zero
• Three types of row operations transform matrix into row echelon form:
  1. Swap positions of two rows
  2. Multiply a row by a non-zero constant
  3. Add a multiple of one row to another row
• Transforming a matrix into row echelon form:
  1. Use row operations to make first non-zero entry in first column equal to 1 (pivot)
  2. Use row operations to make all entries below pivot in first column zero
  3. Repeat process for next column, starting with next row down
  4. Continue until matrix is in row echelon form

Interpreting row echelon form

• Row echelon form of augmented matrix determines solution of linear system
• Types of solutions:
  • Unique solution: Single solution set
    • Pivot in every column (except last) in row echelon form
  • Infinitely many solutions: Infinite number of solution sets
    • At least one free variable (column without pivot) in row echelon form
  • No solution: No solution set
    • Row where all entries are zero except last column in row echelon form

Gaussian elimination for systems

• Method for solving systems of linear equations using row operations
• Transforms augmented matrix into row echelon form
• Steps for Gaussian elimination:
  1. Write system of equations as augmented matrix
  2. Use row operations to transform matrix into row echelon form
  3. Interpret row echelon form to determine solution of system
• Systems with two variables:
  • Row echelon form has two rows
  • Solution found by solving for one variable in terms of other (if unique solution)
• Systems with three variables:
  • Row echelon form has three rows
  • Solution found by solving for one variable in terms of others (if unique solution)
  • If infinitely many solutions, express solution in terms of free variable(s)

Linear Systems and Matrices

• A linear system is a set of equations where each variable appears to the first power
• A matrix is a rectangular array of numbers used to represent a linear system
• Variables in a linear system are the unknowns to be solved for