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!