Math Problem Statement

{ 2x_1 -+x_2 + x_3 = 2 -x_1 +x_2 + -2х_3 = -3 x_1 - x_2 +-2x_3 = -5 методом обратной матрицы

Solution

We are given the system of equations:

2x1+x2+x3=22x_1 + x_2 + x_3 = 2 x1+x22x3=3-x_1 + x_2 - 2x_3 = -3 x1x22x3=5x_1 - x_2 - 2x_3 = -5

This system can be solved using the method of inverse matrices.

Step 1: Write the system as AX=BA \cdot X = B

We can express this system in matrix form as:

2 & 1 & 1 \\ -1 & 1 & -2 \\ 1 & -1 & -2 \end{bmatrix} \cdot \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 2 \\ -3 \\ -5 \end{bmatrix}$$ Where: - $$A$$ is the coefficient matrix: $$A = \begin{bmatrix} 2 & 1 & 1 \\ -1 & 1 & -2 \\ 1 & -1 & -2 \end{bmatrix}$$ - $$X$$ is the vector of unknowns: $$X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$ - $$B$$ is the constants vector: $$B = \begin{bmatrix} 2 \\ -3 \\ -5 \end{bmatrix}$$ ### Step 2: Find the inverse of matrix $$A$$ To solve for $$X$$, we need to compute the inverse of matrix $$A$$, if it exists. The solution to the system is: $$X = A^{-1} \cdot B$$ ### Step 3: Calculate $$A^{-1}$$ We will first find the determinant of matrix $$A$$, and if it is non-zero, we can proceed to compute $$A^{-1}$$. ### Step 4: Multiply $$A^{-1}$$ with $$B$$ to find $$X$$ Once we have $$A^{-1}$$, we can multiply it by $$B$$ to obtain the values for $$x_1$$, $$x_2$$, and $$x_3$$. Let me calculate this for you. numpy as np # Define the coefficient matrix A and the constant matrix B A = np.array([[2, 1, 1], [-1, 1, -2], [1, -1, -2]]) B = np.array([2, -3, -5]) # Calculate the inverse of matrix A A_inv = np.linalg.inv(A) # Solve for X = A_inv * B X = np.dot(A_inv, B) XThe solution to the system of equations is: $$x_1 = -\frac{1}{3}, \quad x_2 = \frac{2}{3}, \quad x_3 = 2$$ If you would like further details or have any questions, feel free to ask! ### Related questions: 1. How do you calculate the determinant of a matrix to check if it is invertible? 2. How do you find the inverse of a 3x3 matrix manually? 3. What happens if the matrix $$A$$ is singular (i.e., non-invertible)? 4. How do you use row reduction to solve systems of linear equations instead of the inverse matrix method? 5. What are some practical applications of solving systems of equations using matrices? **Tip**: Always check the determinant of a matrix before attempting to find its inverse. If the determinant is zero, the matrix does not have an inverse, and the system may be inconsistent or have infinitely many solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Inversion

Formulas

Matrix representation: A · X = B
Inverse matrix formula: X = A^-1 · B
Determinant calculation for invertibility

Theorems

Matrix Inversion Theorem
Determinant and Invertibility Theorem

Suitable Grade Level

Grades 11-12 or College Level