Grade 11 Maths Unit 3 Part 5 Reduced Row Echelon Form of a matrix & System of linear Equations.....

Introduction

This tutorial covers the Reduced Row Echelon Form (RREF) of a matrix and how to solve systems of linear equations with two or three variables. Understanding RREF is essential for simplifying matrices and solving equations effectively in Grade 11 Maths.

Step 1: Understand Row Echelon Form

• Definition: Row Echelon Form (REF) is a matrix form where:

  • All non-zero rows are above any zero rows.
  • Each leading entry of a non-zero row is to the right of the leading entry of the previous row.
  • All entries in a column below a leading entry are zeros.

• Practical Tip: To convert a matrix to REF, use elementary row operations:

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

Step 2: Transition to Reduced Row Echelon Form

• Definition: Reduced Row Echelon Form (RREF) is achieved from REF by ensuring:

  • Each leading entry is 1 (called a leading 1).
  • Each leading 1 is the only non-zero entry in its column.

• Practical Steps:

  1. Start with the matrix in REF.
  2. Scale the rows to ensure each leading entry is 1.
  3. Use row operations to eliminate all other entries in the leading 1's column.

Step 3: Solve Systems of Linear Equations

• Setting Up the Matrix: Represent the system of equations in augmented matrix form. For example, for the equations:

  • 2x + 3y = 5
  • 4x + y = 11 The augmented matrix will be:

  [ 2  3 |  5 ]
  [ 4  1 | 11 ]
  

• Converting to RREF:

  • Use row operations to convert the augmented matrix to RREF.
  • Example steps:
    1. Multiply the first row by 1/2 to get a leading 1:

    [ 1  3/2 | 5/2 ]
    [ 4  1 | 11 ]
    

    2. Subtract 4 times the first row from the second row to eliminate the leading coefficient:

    [ 1  3/2 |  5/2 ]
    [ 0 -5 |  1 ]
    

Step 4: Interpret the Results

• Once in RREF, read the matrix to find the values of the variables:
  • If the matrix is consistent (has a solution), the last column will provide the solutions directly.
  • If there are free variables or inconsistencies (like a row of zeros equating to a non-zero number), identify the nature of the solution (unique, infinite, or none).

Conclusion

Understanding the process of converting a matrix to RREF and solving systems of linear equations is crucial in Grade 11 Maths. Practice with different sets of equations to become proficient. As a next step, try solving a variety of systems using the techniques outlined in this guide to reinforce your understanding.