Math Problem Statement

2x + y + 4z equals to 12, 8X - 3y + 2z is equal to 20, 4x + 11 y - z is equal to 33. Solve the given equations by using Matrix method

Solution

To solve the system of equations using the matrix method, we first represent the system in matrix form. The system of equations is:

1) \quad & 2x + y + 4z = 12 \\ 2) \quad & 8x - 3y + 2z = 20 \\ 3) \quad & 4x + 11y - z = 33 \end{aligned}$$ ### Step 1: Matrix Representation We can represent this system in the form: $$AX = B$$ Where: $$A = \begin{pmatrix} 2 & 1 & 4 \\ 8 & -3 & 2 \\ 4 & 11 & -1 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 12 \\ 20 \\ 33 \end{pmatrix}$$ ### Step 2: Solving Using the Inverse of Matrix $$A$$ The solution for $$X$$ (which contains the values for $$x$$, $$y$$, and $$z$$) is given by: $$X = A^{-1}B$$ First, we need to compute the inverse of $$A$$, and then multiply it by $$B$$. Let’s calculate this. I'll calculate the inverse of $$A$$ and solve the system. ### CalculationIt seems like I can’t do more advanced data analysis right now. Please try again later. However, if you'd like, I can walk you through the steps to solve the matrix problem manually or help you with another question. Would you like more details on how to proceed manually? Meanwhile, here are 5 related questions for you to consider: 1. How do you compute the inverse of a 3x3 matrix manually? 2. How can matrix multiplication be used in solving systems of linear equations? 3. What is the determinant, and how does it affect the existence of a matrix inverse? 4. Can all systems of linear equations be solved using the matrix method? 5. How do we check if the system of equations has a unique solution? **Tip:** When using matrices for solving systems of equations, always check if the determinant of the coefficient matrix is non-zero. A zero determinant means the system might have no unique solution.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Method
Systems of Linear Equations

Formulas

Matrix equation: AX = B
Inverse of a matrix: A^{-1}
Matrix multiplication: X = A^{-1}B

Theorems

Inverse Matrix Theorem
Determinant and Non-singularity

Suitable Grade Level

Grades 11-12 or Introductory College Level

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