14) Cramer's Rule

Cramer's Rule - Solving Systems of Equations

Cramer's Rule - Solving Systems of Linear Equations

1. Understanding Cramer's Rule

Cramer’s Rule is used to solve systems of n linear equations with n unknowns using determinants. It applies only when the determinant of the coefficient matrix is nonzero.

2. Step-by-Step Algorithm

Step 1: Compute det(A)

Find the determinant of matrix A. If det(A) = 0, the system has no unique solution.

Step 2: Construct Matrices A_1, A_2, ...

For each variable, replace the corresponding column of A with the right-hand side values (B).

Step 3: Compute Determinants

Find det(A_1), det(A_2), ... for each modified matrix.

Step 4: Compute Solutions

Use the formula:

x_i = det(A_i) / det(A)

3. Real-World Applications

  • Computer Graphics: Used in transformations and rendering.
  • Electrical Circuits: Solving current equations in networks.
  • Economics: Used in equilibrium analysis.

4. Example Calculation

Given the system:

        2x + 3y = 8
        4x - 5y = -2

Matrix form:

        A = | 2  3 |
            | 4 -5 |
        B = | 8 |
            | -2 |

Compute determinants:

        det(A) = (2)(-5) - (3)(4) = -22
        det(A_1) = (8)(-5) - (3)(-2) = -34
        det(A_2) = (2)(-2) - (8)(4) = -36

Solutions:

        x = -34 / -22 = 17/11
        y = -36 / -22 = 18/11

5. Advantages & Limitations

  • Advantages: Exact solutions, useful for small systems.
  • Limitations: Computationally expensive for large systems, fails when det(A) = 0.

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