Solving a System of Linear Equations using the Inverse Matrix Method

Math Problem Statement

Solve the following system by the inverse matrix method, if possible. If the inverse matrix method doesn't apply, use any other method to determine if the system is inconsistent or dependent. If there is a solution, write your answer in the format (x, y, z). $-2y - 2z = -2$ $-x - y + 3z = 26$ $-3x + 5y + 8z = 32$

Solution

The problem provided involves solving a system of linear equations using the inverse matrix method, or an alternative approach if the inverse matrix method cannot be applied. Let me solve it step by step.

The system of equations is:

$-2y - 2z = -2$
$-x - y + 3z = 26$
$-3x + 5y + 8z = 32$

Step 1: Represent the system in matrix form

The matrix equation $AX = B$ represents the system, where:

\begin{bmatrix} 0 & -2 & -2 \\ -1 & -1 & 3 \\ -3 & 5 & 8 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} -2 \\ 26 \\ 32 \end{bmatrix}.$$ ### Step 2: Check if the inverse of $$A$$ exists To use the inverse matrix method, the determinant of $$A$$ must not be zero. I will calculate the determinant of $$A$$. $$\text{det}(A) = \begin{vmatrix} 0 & -2 & -2 \\ -1 & -1 & 3 \\ -3 & 5 & 8 \end{vmatrix}$$ Let’s compute this determinant. numpy as np # Define matrix A and vector B A = np.array([[0, -2, -2], [-1, -1, 3], [-3, 5, 8]]) B = np.array([-2, 26, 32]) # Calculate the determinant of A det_A = np.linalg.det(A) det_AThe determinant of matrix $$A$$ is $$\text{det}(A) = 18$$, which is nonzero. Therefore, the inverse matrix method can be applied. ### Step 3: Calculate the inverse of $$A$$ Next, I will compute the inverse of $$A$$. Then, I will solve for $$X = A^{-1}B$$. Calculate the inverse of A A_inv = np.linalg.inv(A) # Solve for X = A_inv * B X = np.dot(A_inv, B) XThe solution to the system of equations is: $$(x, y, z) = (-3, -5, 6)$$ ### Final Answer: - There is **one solution**: $$(-3, -5, 6)$$. Would you like a detailed breakdown of each calculation? Here are 5 related questions to expand your understanding: 1. How is the determinant of a matrix calculated, and why is it important? 2. What happens if the determinant of $$A$$ is zero in such problems? 3. How does the inverse matrix method compare to Gaussian elimination? 4. Can this system be solved graphically, and if so, how? 5. What are the implications of dependent or inconsistent systems? **Tip**: Always verify the determinant before attempting to use the inverse matrix method, as it ensures the matrix is invertible!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Inverse Matrix Method

Formulas

Matrix Equation: AX = B
Determinant of a Matrix
Inverse of a Matrix
Matrix Multiplication: X = A^{-1}B

Theorems

Determinant Theorem: A matrix is invertible if and only if its determinant is nonzero.

Suitable Grade Level

Grades 10-12

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