Math Problem Statement
Use matrix inversion to solve the given system of linear equations: x + 2y = -2, y - z = 0, x + 3y - 2z = 1.
Solution
The system of linear equations provided in the image is:
x + 2y &= -2 \\ y - z &= 0 \\ x + 3y - 2z &= 1 \end{aligned}$$ We need to solve this using matrix inversion. ### Step 1: Write the system in matrix form The system of equations can be written as: $$A \cdot \mathbf{x} = \mathbf{b}$$ Where: - $$A$$ is the coefficient matrix - $$\mathbf{x}$$ is the vector of variables $$[x, y, z]$$ - $$\mathbf{b}$$ is the constant matrix For this system, the coefficient matrix $$A$$ and the constant matrix $$\mathbf{b}$$ are: $$A = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 1 & -1 \\ 1 & 3 & -2 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -2 \\ 0 \\ 1 \end{pmatrix}$$ ### Step 2: Find the inverse of matrix $$A$$ We can compute the inverse of $$A$$ (denoted as $$A^{-1}$$) using standard methods such as the adjoint formula or row reduction, but for now, let me calculate the inverse directly: $$A^{-1} = \text{(inverse of A)}$$ ### Step 3: Multiply $$A^{-1}$$ by $$\mathbf{b}$$ Once we have the inverse of $$A$$, we can find the solution vector $$\mathbf{x}$$ by multiplying $$A^{-1}$$ by $$\mathbf{b}$$: $$\mathbf{x} = A^{-1} \cdot \mathbf{b}$$ Let me calculate the inverse of $$A$$ and the solution for $$\mathbf{x}$$.The solution to the system of linear equations using matrix inversion is: $$(x, y, z) = (4, -3, -3)$$ Would you like further details on any of these steps? Here are some related questions to explore this topic further: 1. How do you calculate the inverse of a matrix manually? 2. What conditions make a matrix invertible? 3. Can matrix inversion be used for all systems of linear equations? 4. How does matrix inversion differ from Gaussian elimination? 5. What are some real-world applications of solving systems of equations using matrix methods? **Tip**: When dealing with large matrices, it's often more efficient to use methods like LU decomposition or other factorizations instead of directly computing the inverse!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Systems of Linear Equations
Formulas
Matrix form: A · x = b
Inverse of a matrix: A⁻¹
Solution: x = A⁻¹ · b
Theorems
Matrix Inversion Theorem
Suitable Grade Level
Undergraduate level or advanced high school