Math Problem Statement
Решить систему уравнений методом обратной матрицы
2x1 - x2 + 3x3 = 7
x1 + 3x2 - 2x3 = 0
2x2 - x3 = 2
Solution
The image shows a system of linear equations written in Russian, with a request to solve it using the method of the inverse matrix.
The system is:
2x_1 - x_2 + 3x_3 &= 7, \\ x_1 + 3x_2 - 2x_3 &= 0, \\ 2x_2 - x_3 &= 2. \end{aligned}$$ To solve this system using the method of the inverse matrix, we first represent the system in matrix form: $$A \cdot X = B,$$ where $$A = \begin{pmatrix} 2 & -1 & 3 \\ 1 & 3 & -2 \\ 0 & 2 & -1 \end{pmatrix}, \quad X = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \quad B = \begin{pmatrix} 7 \\ 0 \\ 2 \end{pmatrix}.$$ To find $$X$$, we use the inverse of matrix $$A$$, given by: $$X = A^{-1} \cdot B.$$ I will calculate the inverse of matrix $$A$$ and then multiply it by the vector $$B$$ to find the solution. Let me compute that for you. numpy as np # Defining the matrix A and vector B A = np.array([[2, -1, 3], [1, 3, -2], [0, 2, -1]]) B = np.array([7, 0, 2]) # Computing the inverse of matrix A A_inv = np.linalg.inv(A) # Solving for X using the inverse of A X = np.dot(A_inv, B) XThe solution to the system of equations is: $$x_1 = -1, \quad x_2 = 3, \quad x_3 = 4.$$ Would you like further details on the solution or any explanations about the steps? Here are some related questions you may find interesting: 1. How can I verify the solution to this system of equations? 2. What is the meaning of the inverse matrix method for solving systems? 3. How do I find the determinant of a matrix, and why is it important for invertibility? 4. What are the other methods to solve systems of linear equations? 5. How do I perform matrix multiplication manually for systems of equations? Tip: The inverse matrix method can be computationally expensive for larger systems; in those cases, other methods like Gaussian elimination might be more efficient.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Systems of Linear Equations
Formulas
A * X = B
X = A^-1 * B
Theorems
Inverse Matrix Theorem
Matrix Multiplication
Suitable Grade Level
Grades 10-12
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