Math Problem Statement

Write the system of linear equations in the form Ax = b and solve this matrix equation for x. 2x1 − x2 + x4 = 17 3x2 − x3 − x4 = −17 x1 + x3 − 3x4 = −6 x1 + x2 + 2x3 = 0

Solution

We are given the following system of linear equations:

2x_1 - x_2 + x_4 &= 17 \quad \text{(1)} \\ 3x_2 - x_3 - x_4 &= -17 \quad \text{(2)} \\ x_1 + x_3 - 3x_4 &= -6 \quad \text{(3)} \\ x_1 + x_2 + 2x_3 &= 0 \quad \text{(4)} \end{aligned}$$ ### Step 1: Write in matrix form $$Ax = b$$ We will write this system in matrix form $$Ax = b$$, where: - $$A$$ is the coefficient matrix. - $$x$$ is the column vector of variables. - $$b$$ is the column vector of constants. We can express the system as: $$\begin{pmatrix} 2 & -1 & 0 & 1 \\ 0 & 3 & -1 & -1 \\ 1 & 0 & 1 & -3 \\ 1 & 1 & 2 & 0 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} 17 \\ -17 \\ -6 \\ 0 \end{pmatrix}$$ Thus, the matrix equation $$Ax = b$$ is: $$A = \begin{pmatrix} 2 & -1 & 0 & 1 \\ 0 & 3 & -1 & -1 \\ 1 & 0 & 1 & -3 \\ 1 & 1 & 2 & 0 \end{pmatrix}, \quad x = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix}, \quad b = \begin{pmatrix} 17 \\ -17 \\ -6 \\ 0 \end{pmatrix}$$ ### Step 2: Solve the matrix equation $$Ax = b$$ We can solve this equation by finding the inverse of $$A$$, if it exists, and then multiplying it by $$b$$. The solution is given by: $$x = A^{-1}b$$ Let's proceed to solve the equation by finding the inverse of matrix $$A$$. numpy as np # Define the coefficient matrix A and the constant vector b A = np.array([[2, -1, 0, 1], [0, 3, -1, -1], [1, 0, 1, -3], [1, 1, 2, 0]]) b = np.array([17, -17, -6, 0]) # Calculate the inverse of matrix A A_inv = np.linalg.inv(A) # Solve for x x = np.dot(A_inv, b) xThe solution to the system of equations is: $$x_1 = 4.5, \quad x_2 = -4.5, \quad x_3 = 0, \quad x_4 = 3.5$$ ### Would you like further clarification or more steps in solving this problem? Here are 5 related questions you might find helpful: 1. How can we verify this solution by substituting it back into the original system of equations? 2. What are the conditions under which a matrix $$A$$ has an inverse? 3. How do we perform matrix row reduction to solve a system of linear equations? 4. What is the geometric interpretation of solving a system of linear equations? 5. How can we solve systems of equations using Gaussian elimination? **Tip:** Always check if the determinant of the matrix is non-zero before attempting to find the inverse. If the determinant is zero, the matrix is singular, and the system may have no solution or infinitely many solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Operations
Inverse Matrices

Formulas

Ax = b
x = A^{-1}b

Theorems

Inverse of a matrix
Solution to a system of linear equations using matrices

Suitable Grade Level

Grades 10-12

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