Assignment Gauss Jordan elimination method

Introduction

This tutorial covers the Gauss-Jordan elimination method, a systematic approach used to solve systems of linear equations. Understanding this method is crucial for students in mathematics and engineering, as it provides a clear pathway to obtain solutions for complex systems.

Step 1: Set Up the Augmented Matrix

• Identify the System of Equations: Begin by writing down the system of equations you want to solve.

• Construct the Augmented Matrix: Convert the equations into an augmented matrix form, where each row represents an equation and each column corresponds to the coefficients of the variables. The last column includes the constants from the right side of the equations.

  Example: For the equations:

  2x + 3y = 5
  4x - y = 1
  

  The augmented matrix would be:

  [ 2  3 | 5 ]
  [ 4 -1 | 1 ]
  

Step 2: Apply Row Operations

• Row Operations: Use the following operations to manipulate the rows of the matrix:

  1. Swap two rows.
  2. Multiply a row by a non-zero scalar.
  3. Add or subtract a multiple of one row from another row.

• Goal: Your aim is to convert the matrix into Reduced Row Echelon Form (RREF).

Step 3: Achieve Leading 1s

• First Row: Start with the first row and ensure the leading coefficient (the first non-zero number from the left) is 1. If it’s not, divide the entire row by that coefficient.
• Zero Out Below: Use row operations to create zeros below the leading 1 by subtracting appropriate multiples of the first row from the rows below.

Step 4: Continue to the Next Rows

• Move to the Next Row: Repeat the process for the second row. Ensure it has a leading 1 and zero out all entries below it.
• Proceed Sequentially: Continue this process for all rows, moving from left to right and top to bottom.

Step 5: Back Substitution

• Resulting Matrix: Once in RREF, the matrix should have leading 1s with zeros elsewhere.
• Interpret the Solution: Each leading 1 corresponds to a variable, and the column indicates the value of that variable. If a row corresponds to something like 0 = 1, the system has no solution. If you get a row of zeros, it indicates infinite solutions.

Conclusion

The Gauss-Jordan elimination method allows you to systematically solve systems of equations, providing clear solutions or insights into the nature of the system. To practice, try solving various systems with different numbers of equations and variables. Understanding this method will be beneficial in higher-level mathematics and practical applications in engineering and science.