Math Problem Statement

Use a matrix method to solve: 9y + 2z = 18 3x + 2y + z = 1 x - y = - 3

Solution

To solve the system of linear equations using a matrix method, we can use the method of matrix inversion or Gaussian elimination. Let's start by expressing the system in matrix form:

Given system:

9y + 2z &= 18 \quad \text{(Equation 1)} \\ 3x + 2y + z &= 1 \quad \text{(Equation 2)} \\ x - y &= -3 \quad \text{(Equation 3)} \end{aligned}$$ ### Step 1: Express the system in matrix form We write the system as $$AX = B$$, where $$A$$ is the coefficient matrix, $$X$$ is the vector of variables, and $$B$$ is the constant matrix: $$A = \begin{pmatrix} 0 & 9 & 2 \\ 3 & 2 & 1 \\ 1 & -1 & 0 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 18 \\ 1 \\ -3 \end{pmatrix}$$ So, the system becomes: $$\begin{pmatrix} 0 & 9 & 2 \\ 3 & 2 & 1 \\ 1 & -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 18 \\ 1 \\ -3 \end{pmatrix}$$ ### Step 2: Solve using matrix inversion (if the inverse exists) To solve for $$X$$, we compute: $$X = A^{-1}B$$ Let's calculate the inverse of matrix $$A$$. $$A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)$$ Where $$\text{det}(A)$$ is the determinant of $$A$$ and $$\text{adj}(A)$$ is the adjugate of $$A$$. Let's compute this step by step. ### Step 3: Calculate the determinant of $$A$$ $$\text{det}(A) = 0\cdot(2\cdot0 - 1\cdot(-1)) - 9\cdot(3\cdot0 - 1\cdot1) + 2\cdot(3\cdot(-1) - 2\cdot1)$$ $$\text{det}(A) = 0 - 9\cdot(-1) + 2\cdot(-3 - 2)$$ $$\text{det}(A) = 9 + 2\cdot(-5) = 9 - 10 = -1$$ ### Step 4: Find the adjugate matrix $$\text{adj}(A)$$ This involves finding the cofactor matrix of $$A$$ and then transposing it. Let's compute it. ### Step 5: Multiply $$A^{-1}$$ with $$B$$ to find $$X$$ $$X = A^{-1}B$$ Now I'll calculate each step and find the solution.### Step 6: Results 1. **Determinant of $$A$$**: The determinant of matrix $$A$$ is $$-1$$, which is non-zero, so $$A$$ is invertible. 2. **Inverse of $$A$$**: $$A^{-1} = \begin{pmatrix} -1 & 2 & -5 \\ -1 & 2 & -6 \\ 5 & -9 & 27 \end{pmatrix}$$ 3. **Solution vector $$X$$**: $$X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \\ 0 \end{pmatrix}$$ ### Final Answer: - $$x = -1$$ - $$y = 2$$ - $$z = 0$$ Would you like more details on this method or have any other questions? Here are 5 additional questions you might find helpful: 1. How do you interpret the determinant of a matrix in the context of linear equations? 2. What is the significance of a matrix being invertible? 3. How can you verify the solution using substitution? 4. Can you solve the system using Gaussian elimination instead of matrix inversion? 5. What are some real-world applications of solving linear systems with matrices? **Tip:** Always check the determinant first before attempting to find the inverse of a matrix, as a zero determinant indicates that the matrix is singular and does not have an inverse.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Matrix Operations
Matrix Inversion
Gaussian Elimination

Formulas

Matrix representation of linear equations
Matrix inversion formula

Theorems

Existence of inverse for non-singular matrices

Suitable Grade Level

Advanced High School and Above

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