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Найти решения системы уравнений а) по формулам Крамера; б) с помощью обратной матрицы (матричным методом).



# Coefficients of the system of equations
A = np.array([
    [2, -1, 3],
    [1, 3, -1],
    [1, -2, 2]
])

# Constants on the right-hand side
B = np.array([-4, 11, -7])

# Solve using Cramer's rule
det_A = np.linalg.det(A)

if det_A != 0:
    x1_numer = np.linalg.det(np.column_stack((B, A[:, 1], A[:, 2])))
    x2_numer = np.linalg.det(np.column_stack((A[:, 0], B, A[:, 2])))
    x3_numer = np.linalg.det(np.column_stack((A[:, 0], A[:, 1], B)))

    x1_cramer = x1_numer / det_A
    x2_cramer = x2_numer / det_A
    x3_cramer = x3_numer / det_A
else:
    x1_cramer = x2_cramer = x3_cramer = None

# Solve using the inverse matrix method
try:
    A_inv = np.linalg.inv(A)
    solution_matrix_method = np.dot(A_inv, B)
except np.linalg.LinAlgError:
    solution_matrix_method = None

xHere are the solutions to the system of equations:

### a) Using Cramer's Rule:
- \(x_1 = 1\)
- \(x_2 = 3\)
- \(x_3 = -1\)

### b) Using the Matrix Method (Inverse Matrix):
- \(x_1 = 1\)
- \(x_2 = 3\)
- \(x_3 = -1\)

Both methods give the same solution.

Would you like further clarification on the methods used or detailed step-by-step explanations?

Here are 5 related questions for further exploration:
1. How is Cramer's Rule derived from determinants?
2. What are the conditions for using the inverse matrix method?
3. How can we check if a system of equations has no solution?
4. What is the computational complexity difference between these two methods?
5. How can these methods be extended to larger systems of equations?

**Tip:** Always check the determinant of the coefficient matrix to ensure the system has a unique solution before applying Cramer’s Rule or the matrix method!

Learn to solve a system of linear equations using Cramer's Rule and the Matrix Inversion method. Explore key concepts in linear algebra, such as determinants and inverse matrices, with step-by-step solutions for a 3x3 system of equations. Suitable for advanced high school or undergraduate students.

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