Math Problem Statement

  1. Solve the following system of linear equations using the matrix method

2x+3y-z = 5

4x-y+2z=6

-2x+y+3z = 4

Solution

To solve the system of linear equations using the matrix method, we'll follow these steps:

Step 1: Write the system in matrix form

The system of linear equations is:

2x + 3y - z &= 5 \\ 4x - y + 2z &= 6 \\ -2x + y + 3z &= 4 \end{aligned}$$ We can represent this system as a matrix equation of the form $$AX = B$$, where: $$A = \begin{pmatrix} 2 & 3 & -1 \\ 4 & -1 & 2 \\ -2 & 1 & 3 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 5 \\ 6 \\ 4 \end{pmatrix}$$ ### Step 2: Find the inverse of matrix $$A$$ (if it exists) To solve for $$X$$, we need to find the inverse of matrix $$A$$ and then multiply it by matrix $$B$$. The inverse of a 3x3 matrix $$A$$ is given by: $$A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)$$ where $$\text{det}(A)$$ is the determinant of $$A$$ and $$\text{adj}(A)$$ is the adjugate (or adjoint) of $$A$$. ### Step 3: Calculate the determinant of matrix $$A$$ Let me calculate the determinant of $$A$$, the adjugate of $$A$$, and then solve for $$X$$.The determinant of matrix $$A$$ is approximately $$32.0$$, which is non-zero. This means that the matrix $$A$$ is invertible, and we can find the solution for $$X$$ using the matrix method. The solution vector $$X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$ is approximately: $$X = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$$ ### Final Answer: $$x = 1, \quad y = 2, \quad z = -1$$ Would you like further details on this solution, or do you have any other questions? --- **Related Questions:** 1. How do you find the adjugate of a matrix? 2. What are the properties of matrix determinants? 3. How can you solve a system of equations if the determinant is zero? 4. What is the geometric interpretation of a system of linear equations? 5. How do you interpret the results of a system of linear equations in real-world applications? **Tip:** When working with matrices, always check the determinant first to know whether the matrix is invertible.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Matrix Operations
Determinants
Inverses

Formulas

Matrix equation AX = B
Inverse of a 3x3 matrix

Theorems

Existence of Solutions in Matrix Equations

Suitable Grade Level

High School