Math Problem Statement

Use matrix inversion to solve the given system of linear equations. (You previously solved this system using row reduction.) −x + 2y − z = 0 −x − y + 2z = 0 2x − z = 2

Solution

To solve the given system of linear equations using matrix inversion, we first represent the system in matrix form:

-1 & 2 & -1 \\ -1 & -1 & 2 \\ 2 & 0 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 2 \end{pmatrix}$$ This can be written as: $$A \vec{x} = \vec{b}$$ Where: - $$A$$ is the coefficient matrix: $$A = \begin{pmatrix} -1 & 2 & -1 \\ -1 & -1 & 2 \\ 2 & 0 & -1 \end{pmatrix}$$ - $$\vec{x}$$ is the column vector of variables: $$\vec{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$ - $$\vec{b}$$ is the constant vector: $$\vec{b} = \begin{pmatrix} 0 \\ 0 \\ 2 \end{pmatrix}$$ ### Step 1: Find the inverse of matrix $$A$$ The solution to the system is given by: $$\vec{x} = A^{-1} \vec{b}$$ Let's calculate the inverse of matrix $$A$$. $$A = \begin{pmatrix} -1 & 2 & -1 \\ -1 & -1 & 2 \\ 2 & 0 & -1 \end{pmatrix}$$ We'll find $$A^{-1}$$ and then multiply it by $$\vec{b}$$ to find $$\vec{x}$$. ### Step 2: Calculate $$\vec{x}$$ using matrix inversion Now, I will calculate the inverse of matrix $$A$$ and solve the system.The solution to the system of linear equations is: $$x = 2, \quad y = 2, \quad z = 2$$ This solution was obtained by using matrix inversion, where $$A^{-1}$$ was multiplied by the constant vector $$\vec{b}$$. Would you like further details on the process, or do you have any other questions? Here are 5 related questions: 1. What is the determinant of matrix $$A$$, and why is it important in finding the inverse? 2. How does matrix inversion compare to row reduction when solving systems of equations? 3. Can all systems of linear equations be solved using matrix inversion? 4. What are the properties of an invertible matrix? 5. How do we verify if the computed solution is correct? **Tip:** The determinant of a matrix tells you whether the matrix is invertible. If the determinant is zero, the matrix 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: A * x = b
Inverse of matrix: A^(-1)
Solution: x = A^(-1) * b

Theorems

Invertible Matrix Theorem
Determinant for Matrix Inversion

Suitable Grade Level

Undergraduate Level

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