Lecture Notes on Matrices and Solving Systems of Linear Equations

Introduction to Matrices

• Understanding how to represent systems of linear equations with matrices.
• Goal: Determine the number of solutions (zero, one, or infinitely many) and find the exact solution if it exists.

Techniques for Manipulating Matrices

Main Strategy

• Manipulating the system to create a new, simpler system of equations.
• Valid algebraic manipulations can be applied to both sides of the equation.

Example of a System of Equations

• Given two linear equations:
  1. 4x + 6y = -2
  2. 5x - 3y = 8

Creating the Augmented Matrix

• Corresponding augmented matrix is a 2x3 matrix:
  | 4 | 6 | -2 |
  | 5 | -3 | 8 |

Elementary Row Operations

Techniques Available

1. Multiplying or Dividing Rows

   • Example: Divide the first row by 2.
     • New first row: 2x + 3y = -1

2. Adding or Subtracting Rows

   • Example: Add the first row to the second row.
     • New second row: 7 | 0 | 7 (interpreted as 7x + 0y = 7)
     • Simplifies to x = 1, then find y = -1.

Gauss-Jordan Elimination

• Goal: Achieve a form where each row has a leading 1 and zeros elsewhere, except for the rightmost column.
• Process to solve systems is called Gauss-Jordan elimination.

Full Example of Gauss-Jordan Elimination

Given equations:

1. x + 2y + 3z = 1
2. 2x + 4y + 7z = 2
3. 3x + 7y + 11z = 8

Augmented Matrix Construction

• Form a 3x4 matrix: | 1 | 2 | 3 | 1 |
  | 2 | 4 | 7 | 2 |
  | 3 | 7 | 11 | 8 |

Performing Row Operations

• Subtract the first row from the other rows to produce zeros.
• Modify rows effectively to isolate variables.
  • Resulting in isolation of z, then y, and finally x.

Final Outcome of the Example

• After row operations, isolate variables:
  • x = -9
  • y = 5
  • z = 0

Determining the Number of Solutions

• Reduced Row Echelon Form:
  • A matrix is in reduced row echelon form if:
    • Any row with non-zero entries has a leading 1.
    • All other entries in the leading one's column are zeros.
• Rank of the Matrix:
  • Rank = number of leading ones.
  • If rank = number of columns in coefficient matrix, there is at most one solution.
  • If rank < number of columns, there are either infinitely many solutions or no solution.

Summary and Conclusion

• Elementary row operations used in algebra apply to matrices in a systematic way.
• As we continue with linear algebra, we will explore more complex manipulations and concepts.

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