How is Cramer's Rule different from Gauss-Jordan elimination?

Cramer's Rule and Gauss-Jordan elimination are both methods for solving systems of linear equations, but they differ in their approach and efficiency.

Cramer's Rule:

• Uses determinants: Cramer's Rule leverages determinants to solve for each variable in the system. It involves calculating the determinant of the coefficient matrix and several other determinants obtained by replacing a column of the coefficient matrix with the constant terms.
• Limited to square systems: Cramer's Rule is only applicable to systems with the same number of equations and variables (i.e., square systems).
• Can be computationally expensive: For larger systems, calculating determinants can become computationally expensive, making Cramer's Rule less efficient than other methods.

Gauss-Jordan Elimination:

• Uses row operations: Gauss-Jordan elimination systematically transforms the augmented matrix representing the system of equations into reduced row echelon form. This involves performing elementary row operations like swapping rows, multiplying rows by a scalar, and adding multiples of one row to another.
• Applicable to any system: Gauss-Jordan elimination works for both square and non-square systems of linear equations.
• Generally more efficient: For larger systems, Gauss-Jordan elimination is typically more efficient than Cramer's Rule due to its systematic approach and lower computational complexity.

Example:

Let's consider the following system of equations:

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

Cramer's Rule:

1. Calculate the determinant of the coefficient matrix:

   | 2  1 |
   | 1 -3 | = (2)(-3) - (1)(1) = -7

2. Calculate the determinant of the matrix obtained by replacing the first column with the constant terms:

   | 5  1 |
   | -1 -3 | = (5)(-3) - (1)(-1) = -14

3. Calculate the determinant of the matrix obtained by replacing the second column with the constant terms:

   | 2  5 |
   | 1 -1 | = (2)(-1) - (5)(1) = -7

4. Solve for x and y:

   x = det(X) / det(A) = -14 / -7 = 2
   y = det(Y) / det(A) = -7 / -7 = 1

Gauss-Jordan Elimination:

1. Write the augmented matrix:

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

2. Perform row operations to transform the matrix into reduced row echelon form:
   • Divide the first row by 2:

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

   • Subtract the first row from the second row:

     [ 1  1/2 | 5/2 ]
     [ 0 -7/2 | -7/2 ]

   • Multiply the second row by -2/7:

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

   • Subtract 1/2 times the second row from the first row:

     [ 1  0 | 2 ]
     [ 0  1 | 1 ]

3. The solution is x = 2 and y = 1.

Conclusion:

While both Cramer's Rule and Gauss-Jordan elimination solve systems of linear equations, Gauss-Jordan elimination is generally more efficient and applicable to a wider range of systems. Cramer's Rule, however, offers a straightforward approach for solving square systems, especially when dealing with smaller systems.