Math Problem Statement

Решить систему уравнений методом обратной матрицы: 2x1 - x2 + 3x3 = 7, x1 + 3x2 - 2x3 = 0, 2x2 - x3 = 2

Solution

The problem asks you to solve the following system of equations using the method of the inverse matrix:

2x_1 - x_2 + 3x_3 = 7 \\ x_1 + 3x_2 - 2x_3 = 0 \\ 2x_2 - x_3 = 2 \end{cases}$$ To solve this using the inverse matrix method, follow these steps: ### Step 1: Write the system in matrix form $$Ax = b$$ From the given system, you can extract the matrix $$A$$ (the coefficient matrix), $$x$$ (the variable matrix), and $$b$$ (the constant matrix). The system can be written as: $$A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 3 & -2 \\ 0 & 2 & -1 \end{bmatrix}, \quad x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad b = \begin{bmatrix} 7 \\ 0 \\ 2 \end{bmatrix}$$ ### Step 2: Find the inverse of matrix $$A$$ We need to find $$A^{-1}$$, the inverse of matrix $$A$$, to solve for $$x$$. The solution to the system is given by: $$x = A^{-1}b$$ ### Step 3: Compute $$x$$ Once we have the inverse matrix $$A^{-1}$$, we can multiply it by the matrix $$b$$ to obtain the values of $$x_1$$, $$x_2$$, and $$x_3$$. I will now compute the inverse of matrix $$A$$ and solve for $$x$$. numpy as np # Define the matrix A and the matrix b A = np.array([[2, -1, 3], [1, 3, -2], [0, 2, -1]]) b = np.array([7, 0, 2]) # Compute the inverse of A A_inv = np.linalg.inv(A) # Solve for x x = np.dot(A_inv, b) xThe solution to the system of equations is: $$x_1 = -1, \quad x_2 = 3, \quad x_3 = 4$$ Would you like any further explanation or assistance with this solution? Here are some related questions you may find interesting: 1. How do you calculate the determinant of a matrix and why is it important for finding the inverse? 2. What happens if a matrix does not have an inverse? How can you identify such matrices? 3. How can the inverse matrix method be used for larger systems of equations? 4. What is the relationship between matrix multiplication and solving systems of equations? 5. How do you solve systems of equations using Gaussian elimination? **Tip:** Always check if the determinant of the matrix is non-zero before attempting to find its inverse. If the determinant is zero, the matrix is singular and does not have an inverse.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Systems of Linear Equations

Formulas

Matrix form of system: Ax = b
Inverse matrix solution: x = A^(-1)b

Theorems

Properties of inverse matrices
Matrix multiplication rules

Suitable Grade Level

Grades 10-12 or Undergraduate Level

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