Solving System of Linear Equations using Cramer's Rule Principles

 Cramer's rule is a method for solving a system of linear equations using determinants. It's a method that can be applied when the system of equations has the same number of equations as unknowns, and the determinant of the coefficient matrix is not zero.

The system of linear equations can be written in matrix form as Ax = B, where:

- A is the coefficient matrix,

- x is the column vector of unknowns,

- B is the column vector on the right-hand side of the equations.

Cramer's rule expresses the solution for each unknown xi as the ratio of the determinant of a matrix obtained by replacing the corresponding column in the coefficient matrix with the column vector on the right-hand side to the determinant of the original coefficient matrix.

Here are the steps to solve a system of linear equations using Cramer's rule:

1. Write the System of Equations:

   Write the system of linear equations in the form Ax = B.

   Example:

  2x + 3y - z = 1

   4x - y + 2z = -2

   x + 2y + 3z = 3

Can be written in matrix form as Ax = B:

   | 2  3 -1 |   | x |   |  1 |

   | 4 -1  2 | * | y | = | -2 |

   | 1  2  3 |   | z |   |  3 |

2. Calculate the Determinant of the Coefficient Matrix (DetA):

   Calculate the determinant of the coefficient matrix A.

   Example:

   DetA = | 2  3 -1 |

          | 4 -1  2 |

          | 1  2  3 |

3. Calculate the Determinant for Each Variable (DetX, DetY, DetZ):

   Replace the corresponding column of the coefficient matrix with the column vector on the right-hand side to get matrices Ax, Ay, Az, and then calculate their determinants.

   Example:

   DetX = |  1  3 -1 |

          | -2 -1  2 |

          |  3  2  3 |

   

   DetY = | 2   1 -1 |

          | 4  -2  2 |

          | 1   3  3 |

   

   DetZ = | 2  3  1 |

          | 4 -1 -2 |

          | 1  2  3 |

4. Calculate the Solutions:

   Calculate the solutions for each unknown using the following formulas:

   x = DetX / DetA

   y = DetY / DetA

   z = DetZ / DetA

   Example:

   x = DetX / DetA

   y = DetY / DetA

   z = DetZ / DetA

Note: Cramer's rule can be computationally expensive and is not always the most efficient method, especially for large systems of equations. It's more practical for smaller systems or for educational purposes. In practice, numerical methods like Gaussian elimination or matrix factorization methods are often used.