Math Problem Statement
Find solutions to the system of equations a) using Cramer’s formulas; b) using an inverse matrix (matrix method)
Solution
To solve this system of linear equations using the specified methods, let's begin by identifying the equations:
1) & \ 2x_1 - x_2 + 3x_3 = -4, \\ 2) & \ x_1 + 3x_2 - x_3 = 11, \\ 3) & \ x_1 - 2x_2 + 2x_3 = -7. \end{aligned}$$ The methods you’ve requested are: ### (a) Using Cramer’s formulas Cramer's rule states that the solution of the system $$AX = B$$ is given by: $$x_i = \frac{\text{det}(A_i)}{\text{det}(A)},$$ where $$A$$ is the coefficient matrix, $$A_i$$ is $$A$$ with its $$i$$-th column replaced by $$B$$, and $$B$$ is the column of constants. ### (b) Using the matrix inverse The solution is obtained as: $$X = A^{-1}B,$$ where $$A^{-1}$$ is the inverse of the coefficient matrix $$A$$. Let me compute the solutions. numpy as np # Define the coefficient matrix (A) and the constants vector (B) A = np.array([[2, -1, 3], [1, 3, -1], [1, -2, 2]]) B = np.array([-4, 11, -7]) # (a) Using Cramer's Rule det_A = np.linalg.det(A) # Determinant of A # Replace columns of A with B to compute determinants for Cramer's Rule A_1 = A.copy() A_1[:, 0] = B det_A1 = np.linalg.det(A_1) A_2 = A.copy() A_2[:, 1] = B det_A2 = np.linalg.det(A_2) A_3 = A.copy() A_3[:, 2] = B det_A3 = np.linalg.det(A_3) x1_cramer = det_A1 / det_A x2_cramer = det_A2 / det_A x3_cramer = det_A3 / det_A # (b) Using the inverse matrix method A_inv = np.linalg.inv(A) # Compute the inverse of A X_matrix = np.dot(A_inv, B) # Solve for X x1_cramer, x2_cramer, x3_cramer, X_matrixHere are the solutions for the system of equations: ### (a) Using Cramer’s Rule: $$x_1 = 1, \quad x_2 = 3, \quad x_3 = -1.$$ ### (b) Using the Matrix Inverse Method: $$x_1 = 1, \quad x_2 = 3, \quad x_3 = -1.$$ Both methods yield the same solution. Let me know if you need a detailed step-by-step explanation for either approach! --- Here are 5 related questions to deepen your understanding: 1. How is the determinant of a matrix calculated, and what does it signify in Cramer's rule? 2. What are the conditions under which Cramer’s Rule is applicable? 3. How do you interpret the inverse of a matrix in terms of solving systems of equations? 4. What happens if the determinant of the coefficient matrix is zero? 5. How do you solve a system of equations with more than three variables using the same methods? **Tip:** Always verify the determinant of the coefficient matrix when solving using Cramer's rule or the matrix inverse. A determinant of zero indicates the system is either inconsistent or has infinitely many solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Cramer's Rule
Matrix Inversion
Formulas
x_i = det(A_i) / det(A) (Cramer's Rule)
X = A^{-1}B (Matrix Inversion)
Theorems
Cramer's Rule
Inverse Matrix Theorem
Suitable Grade Level
Grades 10-12
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