Math Problem Statement
find x, y and z using inverse matrix method with cofactor x+2y-z=5 3x-y+3z=7 4x-2y+4z=12
Solution
To solve the system of equations using the inverse matrix method with cofactors, let's follow these steps:
Given System of Equations:
Step 1: Write the System in Matrix Form
We can represent the system of equations in matrix form as: Where:
1 & 2 & -1 \\ 3 & -1 & 3 \\ 4 & -2 & 4 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 7 \\ 12 \end{pmatrix}$$ ### Step 2: Find the Determinant of Matrix $$A$$ The determinant of $$A$$ is calculated as: $$\text{det}(A) = 1 \times \begin{vmatrix} -1 & 3 \\ -2 & 4 \end{vmatrix} - 2 \times \begin{vmatrix} 3 & 3 \\ 4 & 4 \end{vmatrix} - 1 \times \begin{vmatrix} 3 & -1 \\ 4 & -2 \end{vmatrix}$$ Let's compute each minor determinant: $$\begin{vmatrix} -1 & 3 \\ -2 & 4 \end{vmatrix} = (-1)(4) - (3)(-2) = -4 + 6 = 2$$ $$\begin{vmatrix} 3 & 3 \\ 4 & 4 \end{vmatrix} = (3)(4) - (3)(4) = 12 - 12 = 0$$ $$\begin{vmatrix} 3 & -1 \\ 4 & -2 \end{vmatrix} = (3)(-2) - (-1)(4) = -6 + 4 = -2$$ So, the determinant is: $$\text{det}(A) = 1(2) - 2(0) - 1(-2) = 2 + 2 = 4$$ ### Step 3: Find the Cofactor Matrix The cofactor matrix $$C$$ is formed by the cofactors of each element in matrix $$A$$. $$C = \begin{pmatrix} \text{C}_{11} & \text{C}_{12} & \text{C}_{13} \\ \text{C}_{21} & \text{C}_{22} & \text{C}_{23} \\ \text{C}_{31} & \text{C}_{32} & \text{C}_{33} \end{pmatrix}$$ Using the minors calculated earlier: $$C = \begin{pmatrix} 2 & 0 & -2 \\ -10 & -4 & -2 \\ 0 & 0 & 0 \end{pmatrix}$$ ### Step 4: Find the Adjugate Matrix $$\text{Adj}(A)$$ The adjugate matrix is the transpose of the cofactor matrix $$C$$: $$\text{Adj}(A) = C^T = \begin{pmatrix} 2 & -10 & 0 \\ 0 & -4 & 0 \\ -2 & -2 & 0 \end{pmatrix}$$ ### Step 5: Find the Inverse Matrix $$A^{-1}$$ The inverse matrix is given by: $$A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{Adj}(A) = \frac{1}{4} \begin{pmatrix} 2 & -10 & 0 \\ 0 & -4 & 0 \\ -2 & -2 & 0 \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & -\frac{5}{2} & 0 \\ 0 & -1 & 0 \\ -\frac{1}{2} & -\frac{1}{2} & 0 \end{pmatrix}$$ ### Step 6: Multiply $$A^{-1}$$ by $$\mathbf{b}$$ to Find $$\mathbf{x}$$ Now, multiply the inverse matrix by $$\mathbf{b}$$: $$\mathbf{x} = A^{-1} \cdot \mathbf{b} = \begin{pmatrix} \frac{1}{2} & -\frac{5}{2} & 0 \\ 0 & -1 & 0 \\ -\frac{1}{2} & -\frac{1}{2} & 0 \end{pmatrix} \cdot \begin{pmatrix} 5 \\ 7 \\ 12 \end{pmatrix}$$ Carrying out the matrix multiplication: $$\mathbf{x} = \begin{pmatrix} \frac{1}{2}(5) + \left(-\frac{5}{2}\right)(7) + 0(12) \\ 0(5) + (-1)(7) + 0(12) \\ -\frac{1}{2}(5) + \left(-\frac{1}{2}\right)(7) + 0(12) \end{pmatrix} = \begin{pmatrix} \frac{5}{2} - \frac{35}{2} \\ -7 \\ -\frac{5}{2} - \frac{7}{2} \end{pmatrix} = \begin{pmatrix} -15 \\ -7 \\ -6 \end{pmatrix}$$ ### Final Answer: $$x = -15, \quad y = -7, \quad z = -6$$ Would you like more details, or do you have any questions about the process? Here are some related questions: 1. What are the steps to calculate the determinant of a 3x3 matrix? 2. How do you calculate the cofactor of a matrix? 3. Why is the inverse of a matrix important in solving systems of equations? 4. Can all matrices be inverted? Why or why not? 5. What does it mean if a matrix has a determinant of zero? **Tip:** Always check the determinant of a matrix before attempting to find its inverse. If the determinant is zero, the matrix is singular, and an inverse does not exist.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Inverse Matrices
Cofactor Expansion
Formulas
Matrix determinant calculation
Inverse matrix formula
Cofactor formula
Theorems
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Suitable Grade Level
Advanced High School
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