Math Problem Statement
solve iii)
Solution
To solve the given system of equations for (iii) using the matrix inverse method, we follow these steps:
The system of equations is:
Step 1: Represent the system in matrix form
This can be written as:
Where:
1 & 1 & 1 \\ 3 & 2 & -1 \\ 3 & 1 & 2 \end{bmatrix}, \quad \mathbf{X} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} 9 \\ 8 \\ 1 \end{bmatrix}$$ ### Step 2: Find the inverse of matrix $$\mathbf{A}$$ The solution is given by: $$\mathbf{X} = \mathbf{A}^{-1} \cdot \mathbf{B}$$ Let us compute the inverse of $$\mathbf{A}$$ and solve for $$\mathbf{X}$$. ### Step 3: Compute the determinant of $$\mathbf{A}$$ The determinant of $$\mathbf{A}$$ is: $$\text{det}(\mathbf{A}) = 1 \cdot (2 \cdot 2 - (-1) \cdot 1) - 1 \cdot (3 \cdot 2 - (-1) \cdot 3) + 1 \cdot (3 \cdot 1 - 3 \cdot 2)$$ $$\text{det}(\mathbf{A}) = 1 \cdot (4 + 1) - 1 \cdot (6 + 3) + 1 \cdot (3 - 6)$$ $$\text{det}(\mathbf{A}) = 5 - 9 - 3 = -7$$ Since $$\text{det}(\mathbf{A}) \neq 0$$, the matrix is invertible. ### Step 4: Compute the inverse of $$\mathbf{A}$$ The formula for the inverse of a 3x3 matrix is: $$\mathbf{A}^{-1} = \frac{1}{\text{det}(\mathbf{A})} \cdot \text{adj}(\mathbf{A})$$ Where $$\text{adj}(\mathbf{A})$$ is the adjugate of $$\mathbf{A}$$, computed using cofactors. #### Cofactors of $$\mathbf{A}$$: 1. For element (1,1): $$\text{Cofactor} = \text{det} \begin{bmatrix} 2 & -1 \\ 1 & 2 \end{bmatrix} = (2)(2) - (-1)(1) = 4 + 1 = 5$$ 2. For element (1,2): $$\text{Cofactor} = -\text{det} \begin{bmatrix} 3 & -1 \\ 3 & 2 \end{bmatrix} = -(3 \cdot 2 - (-1) \cdot 3) = -(6 + 3) = -9$$ 3. For element (1,3): $$\text{Cofactor} = \text{det} \begin{bmatrix} 3 & 2 \\ 3 & 1 \end{bmatrix} = (3)(1) - (2)(3) = 3 - 6 = -3$$ ... Compute for all elements. Let me calculate further. numpy as np # Define matrix A and B A = np.array([ [1, 1, 1], [3, 2, -1], [3, 1, 2] ]) B = np.array([9, 8, 1]) # Compute the inverse of A A_inv = np.linalg.inv(A) # Solve for X X = np.dot(A_inv, B) A_inv, XThe solution to the system of equations is: $$\mathbf{A}^{-1} = \begin{bmatrix} -0.714 & 0.143 & 0.429 \\ 1.286 & 0.143 & -0.571 \\ 0.429 & -0.286 & 0.143 \end{bmatrix}$$ $$\mathbf{X} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -4.857 \\ 12.143 \\ 1.714 \end{bmatrix}$$ Thus: $$x = -4.857, \quad y = 12.143, \quad z = 1.714$$ ### Final Answer: $$x \approx -4.86, \quad y \approx 12.14, \quad z \approx 1.71$$ --- Would you like me to explain any step in further detail? ### Related Questions: 1. How do you calculate the determinant of a 3x3 matrix step by step? 2. What is the geometric interpretation of solving a system of equations using matrix inverses? 3. Can matrix inversion fail for certain systems of equations? If yes, why? 4. How can the computational efficiency of solving systems using matrix inverses be improved? 5. What are the alternative methods to solve systems of equations besides using matrix inverses? ### Tip: Always verify if the determinant of the matrix is non-zero before attempting to calculate its inverse, as only invertible matrices yield unique solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Inversion
Formulas
Matrix form: A · X = B
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Inverse of a 3x3 matrix: A^(-1) = (1/det(A)) · adj(A)
Solution formula: X = A^(-1) · B
Theorems
Matrix Inversion Theorem
Determinants and Invertibility
Suitable Grade Level
Undergraduate - Linear Algebra or Advanced High School Math
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