Math Problem Statement
Solution
Let's solve the given system of linear equations step by step using:
- Cramer's Rule.
- Inverse Matrix Method.
The system of equations is:
x + 2y - 2z &= -1 \\ 2x + 3y &= 5 \\ 2x - 2y + 5z &= 10 \end{aligned}$$ ### (a) **Cramer's Rule**: Cramer's Rule uses determinants to solve a system of linear equations. The system can be represented in matrix form as $$A \mathbf{x} = \mathbf{b}$$, where: $$A = \begin{pmatrix} 1 & 2 & -2 \\ 2 & 3 & 0 \\ 2 & -2 & 5 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -1 \\ 5 \\ 10 \end{pmatrix}$$ The solutions for $$x$$, $$y$$, and $$z$$ are given by: $$x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}, \quad z = \frac{\det(A_z)}{\det(A)}$$ Where $$A_x$$, $$A_y$$, and $$A_z$$ are the matrices formed by replacing the respective columns of $$A$$ with $$\mathbf{b}$$. Let me calculate these determinants for you.For part (a) **Cramer's Rule**, the determinants are: - $$\det(A) = 15$$ - $$\det(A_x) = 15$$ - $$\det(A_y) = 15$$ - $$\det(A_z) = 30$$ Thus, the solution is: $$x = \frac{15}{15} = 1, \quad y = \frac{15}{15} = 1, \quad z = \frac{30}{15} = 2$$ ### (b) **Inverse Matrix Method**: The solution to the system can also be found using the inverse of matrix $$A$$, as follows: $$\mathbf{x} = A^{-1} \mathbf{b}$$ Let me now calculate $$A^{-1}$$ and find the solution.For part (b) **Inverse Matrix Method**, the solution is: $$x = 1, \quad y = 1, \quad z = 2$$ Both methods yield the same result: $$x = 1, \quad y = 1, \quad z = 2$$ Do you want further details on either method or have any questions? Here are 5 related questions: 1. How can Cramer's Rule be applied to larger systems of equations? 2. What is the geometric interpretation of solving a system using matrices? 3. Can Cramer's Rule be used when the determinant of $$A$$ is zero? 4. How is the inverse matrix method affected by ill-conditioned matrices? 5. What are the limitations of using the inverse matrix method for solving systems? **Tip**: Always check the determinant of a matrix before applying Cramer's Rule or the inverse matrix method. If it's zero, the system may not have a unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrix Inversion
Formulas
Cramer's Rule: x = det(A_x) / det(A), y = det(A_y) / det(A), z = det(A_z) / det(A)
Inverse Matrix: x = A^(-1) * b
Theorems
Cramer's Rule: A system of linear equations has a unique solution if det(A) ≠ 0.
Matrix Inversion: A matrix A has an inverse if det(A) ≠ 0.
Suitable Grade Level
College Level (Linear Algebra)
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