Solve System of Equations Using Inverse Matrix

Math Problem Statement

Homework:12.4 HW - Matrix Algebra

Question 30, 12.4.75-GC

HW Score: 87.62%, 30.67 of 35 points

Points: 0 of 1

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Part 1

Use the inverse matrix and matrix multiplication with a graphing utility to solve the following system of equations.

left brace Start 3 By 7 Matrix 1st Row 1st Column 21 x 2nd Column plus 3rd Column 63 y 4st Column minus 5st Column 6 z 6st Column equals 7st Column 16 2nd Row 1st Column 17 x 2nd Column minus 3rd Column 8 y 4st Column plus 5st Column 13 z 6st Column equals 7st Column 9 3rd Row 1st Column 4 x 2nd Column plus 3rd Column 2 y 4st Column minus 5st Column z 6st Column equals 7st Column 12 EndMatrix

21x

63y

−

17x

−

13z

−

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Part 1

xalmost equals≈enter your response here

yalmost equals≈enter your response here

zalmost equals≈enter your response here

(Round to the nearest hundredth as needed.)

xyyzz(Round to the nearest hundredth as needed.)xyyzz(Round to the nearest hundredth as needed.)

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Solution

To solve the system of linear equations using the inverse matrix and matrix multiplication, we can express the system in matrix form as:

$A \cdot \mathbf{x} = \mathbf{b}$

Where:

21 & 63 & -6 \\ 17 & -8 & 13 \\ 4 & 2 & -1 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 16 \\ 9 \\ 12 \end{bmatrix}$$ ### Steps: 1. **Find the inverse of $$A$$, denoted $$A^{-1}$$:** Multiply both sides of the equation by $$A^{-1}$$, yielding: $$\mathbf{x} = A^{-1} \cdot \mathbf{b}$$ 2. **Perform the matrix operations to compute $$A^{-1}$$ and $$A^{-1} \cdot \mathbf{b}$$.** Let's calculate this.import numpy as np # Define matrix A and vector b A = np.array([[21, 63, -6], [17, -8, 13], [4, 2, -1]]) b = np.array([16, 9, 12]) # Calculate the inverse of A A_inv = np.linalg.inv(A) # Solve for x (A^-1 * b) x = np.dot(A_inv, b) xarray([ 2.6372796 , -0.94290512, -3.33669186])array([ 2.6372796 , -0.94290512, -3.33669186])successThe solution to the system of equations is approximately: $$x \approx 2.64, \quad y \approx -0.94, \quad z \approx -3.34$$ ### Final Answer: - $$x = 2.64$$ - $$y = -0.94$$ - $$z = -3.34$$ Would you like further clarification on any part of this calculation? --- **Follow-up questions:** 1. What does the determinant of matrix $$A$$ tell us about the system? 2. How can we verify the solution using substitution? 3. How does rounding affect the accuracy of the solution? 4. What happens if matrix $$A$$ is not invertible? 5. Can we solve this using a different numerical method, such as Gaussian elimination? **Tip:** Always double-check the matrix entries to ensure they match the problem statement exactly when performing manual or computational calculations!

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Linear Equations
Matrix Inverse
Matrix Multiplication

Formulas

A * x = b
x = A^(-1) * b
Inverse of a matrix: A^(-1)

Theorems

Matrix Inversion Theorem
Properties of Matrix Multiplication

Suitable Grade Level

Grades 10-12

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