Math Problem Statement
Using matrix inversion method, solve the system of equations: x + 2y - 6z = 5, 2x - 4y + 5z = 6, 3x - 16y + 9z = 4.
Solution
To solve the given system of equations using the matrix inversion method, follow these steps:
System of Equations:
1. & \quad x + 2y - 6z = 5 \\ 2. & \quad 2x - 4y + 5z = 6 \\ 3. & \quad 3x - 16y + 9z = 4 \end{aligned}$$ ### Step 1: Express the system in matrix form The system can be written as $$A \mathbf{x} = \mathbf{b}$$, where: $$A = \begin{pmatrix} 1 & 2 & -6 \\ 2 & -4 & 5 \\ 3 & -16 & 9 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 6 \\ 4 \end{pmatrix}$$ ### Step 2: Find the inverse of matrix $$A$$ The inverse of a matrix $$A$$, denoted $$A^{-1}$$, is calculated as follows: $$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$$. #### Calculate $$\text{det}(A)$$: $$\text{det}(A) = 1 \times \begin{vmatrix} -4 & 5 \\ -16 & 9 \end{vmatrix} - 2 \times \begin{vmatrix} 2 & 5 \\ 3 & 9 \end{vmatrix} + (-6) \times \begin{vmatrix} 2 & -4 \\ 3 & -16 \end{vmatrix}$$ $$\text{det}(A) = 1 \times ((-4)(9) - (5)(-16)) - 2 \times ((2)(9) - (5)(3)) - 6 \times ((2)(-16) - (-4)(3))$$ $$\text{det}(A) = 1 \times (-36 + 80) - 2 \times (18 - 15) - 6 \times (-32 + 12)$$ $$\text{det}(A) = 1 \times 44 - 2 \times 3 - 6 \times (-20)$$ $$\text{det}(A) = 44 - 6 + 120 = 158$$ #### Calculate $$\text{adj}(A)$$: To calculate the adjugate matrix $$\text{adj}(A)$$, compute the cofactor matrix and then take its transpose. ### Step 3: Compute $$\mathbf{x} = A^{-1} \mathbf{b}$$ Once we have $$A^{-1}$$, multiply it by the vector $$\mathbf{b}$$ to find $$\mathbf{x}$$. Let's perform these calculations step by step. #### Step 2: Finding the Inverse Matrix Let me calculate the inverse matrix $$A^{-1}$$ and then solve for $$\mathbf{x}$$.The solution to the system of equations is: $$x = 4, \quad y = 0.5, \quad z = 0$$ This means the values of $$x$$, $$y$$, and $$z$$ that satisfy all three equations are $$x = 4$$, $$y = 0.5$$, and $$z = 0$$. Would you like more details or have any questions? ### Related Questions: 1. How do you compute the determinant of a 3x3 matrix? 2. What is the adjugate of a matrix, and how is it used in matrix inversion? 3. Can you solve a system of linear equations if the determinant is zero? 4. What are other methods to solve a system of linear equations besides matrix inversion? 5. How does matrix multiplication work in solving systems of equations? 6. What are the applications of matrix inversion in real-world problems? 7. How does numerical precision affect the accuracy of matrix inversion? 8. What is the significance of the inverse matrix in linear algebra? ### Tip: When solving systems of equations using matrices, ensure that the determinant is not zero, as this would mean the matrix is singular and doesn't have an inverse.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Systems of Equations
Formulas
Matrix Inversion Formula
Determinant Calculation
Theorems
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Suitable Grade Level
Advanced High School
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