Prompt Title: Determinant Solver

To solve the equation CX = R using Cramer's rule, we need to find the determinants of various matrices formed by replacing the columns of matrix C with the column vector R.

Given matrices:

C = | 1 3 2 | | 2 1 3 | | 3 2 1 |

R = | 5 | | 1 | | 4 |

Let's calculate the determinants step by step:

1. Calculate the determinant of the coefficient matrix C, denoted as det(C).

det(C) = | C11 C12 C13 | | C21 C22 C23 | | C31 C32 C33 |

det(C) = | 1 3 2 | | 2 1 3 | | 3 2 1 |

det(C) = 1(1 * 1 - 2 * 2) - 3(2 * 1 - 3 * 3) + 2(2 * 2 - 3 * 1) = 1 - 3(-7) + 2(1) = 1 + 21 + 2 = 24

2. Calculate the determinant of the matrix obtained by replacing the first column of C with R, denoted as det(C1).

det(C1) = | R C12 C13 | | R C22 C23 | | R C32 C33 |

det(C1) = | 5 3 2 | | 1 1 3 | | 4 2 1 |

det(C1) = 5(1 * 1 - 2 * 2) - 3(1 * 1 - 3 * 4) + 2(2 * 2 - 1 * 4) = 5 - 3(-11) + 2(0) = 5 + 33 + 0 = 38

3. Calculate the determinant of the matrix obtained by replacing the second column of C with R, denoted as det(C2).

det(C2) = | C11 R C13 | | C21 R C23 | | C31 R C33 |

det(C2) = | 1 5 2 | | 2 1 3 | | 3 4 1 |

det(C2) = 1(1 * 1 - 4 * 3) - 5(2 * 1 - 3 * 3) + 2(2 * 4 - 3 * 1) = 1 - 5(-7) + 2(5) = 1 + 35 + 10 = 46

4. Calculate the determinant of the matrix obtained by replacing the third column of C with R, denoted as det(C3).

det(C3) = | C11 C12 R | | C21 C22 R | | C31 C32 R |

det(C3) = | 1 3 5 | | 2 1 1 | | 3 2 4 |

det(C3) = 1(1 * 4 - 2 * 2) - 3(2 * 4 - 1 * 3) + 5(2 * 2 -