Math Problem Statement
⎛⎝⎜1 1 02 1 20 1 1⎞⎠⎟X=⎛⎝⎜5 −1 2−6 4 6−2 0 7⎞⎠⎟
Solution
We need to solve for in the given matrix equation:
1 & 1 & 0 \\ 2 & 1 & 2 \\ 0 & 1 & 1 \end{bmatrix} X = \begin{bmatrix} 5 & -1 & 2 \\ -6 & 4 & 6 \\ -2 & 0 & 7 \end{bmatrix}.$$ Here, $$X$$ is a matrix, and the equation represents a system of linear equations. The solution for $$X$$ can be found by multiplying the right-hand side matrix by the inverse of the coefficient matrix on the left. ### Steps to Solve: 1. **Define the matrices**: Let $$A$$ be the coefficient matrix: $$A = \begin{bmatrix} 1 & 1 & 0 \\ 2 & 1 & 2 \\ 0 & 1 & 1 \end{bmatrix}.$$ And let $$B$$ be the matrix on the right-hand side: $$B = \begin{bmatrix} 5 & -1 & 2 \\ -6 & 4 & 6 \\ -2 & 0 & 7 \end{bmatrix}.$$ 2. **Find the inverse of $$A$$**: The inverse of $$A$$ is given by $$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$$, where $$\det(A)$$ is the determinant of $$A$$ and $$\text{adj}(A)$$ is the adjugate of $$A$$. 3. **Multiply $$A^{-1} \cdot B$$**: After finding the inverse, multiply it by $$B$$ to obtain $$X$$. Let me compute this for you. numpy as np # Define matrices A and B A = np.array([[1, 1, 0], [2, 1, 2], [0, 1, 1]]) B = np.array([[5, -1, 2], [-6, 4, 6], [-2, 0, 7]]) # Compute the inverse of A A_inv = np.linalg.inv(A) # Compute the solution matrix X X = np.dot(A_inv, B) XThe solution matrix $$X$$ is: $$X = \begin{bmatrix} 1 & 1 & -2 \\ 4 & -2 & 4 \\ -6 & 2 & 3 \end{bmatrix}.$$ ### Verification The result satisfies $$AX = B$$ because we used $$A^{-1}B$$ to calculate $$X$$. Would you like a step-by-step breakdown of the calculations or any further details? Here are some related questions for exploration: 1. What are the properties of matrix inverses and how do they work in solving systems of equations? 2. How can the determinant of a matrix affect the solvability of a system? 3. What happens if $$A$$ is singular (non-invertible)? 4. How can this solution be verified manually using row reduction? 5. What are some real-world applications of solving such matrix equations? **Tip**: Always check the determinant of the matrix $$A$$. If it's zero, the inverse doesn't exist, and you'll need to explore other methods for solving.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Linear Equations
Matrix Inversion
Formulas
Matrix multiplication: (AB)_{ij} = Σ_k A_{ik}B_{kj}
Inverse of a matrix: A^{-1} = (1/det(A)) * adj(A)
Determinant calculation: det(A)
Theorems
Matrix Inversion Theorem
Properties of Determinants
Suitable Grade Level
Grades 11-12 and Undergraduate