Math Problem Statement
Solution
From the image, we are given a system of linear equations:
x + 2y + 2z &= 1 \\ -2x - y + z &= 1 \\ 2x + 2y + 2z &= 4 \end{aligned}$$ 1. **Coefficient matrix (A)** of the system: $$A = \begin{bmatrix} 1 & 2 & 2 \\ -2 & -1 & 1 \\ 2 & 2 & 2 \end{bmatrix}$$ 2. **Task:** Compute the inverse of the matrix $$A$$. Let me compute the inverse matrix for you.The inverse of the matrix $$A$$ is: $$A^{-1} = \begin{bmatrix} -1 & 0 & 1 \\ 1.5 & -0.5 & -1.25 \\ -0.5 & 0.5 & 0.75 \end{bmatrix}$$ ### Now, let's address the statement about whether this system has a unique solution: - A system of linear equations has a unique solution if the coefficient matrix $$A$$ is invertible (i.e., if the determinant of $$A \neq 0$$). - Since the inverse of $$A$$ exists, the determinant is non-zero, meaning the system **does** have a unique solution. Would you like more details on this, or do you have any other questions? Here are some related questions for further exploration: 1. How is the determinant of a matrix calculated, and what does it signify? 2. What are the conditions for a matrix to have an inverse? 3. How can you verify the solution of a system of equations using matrix inversion? 4. What does it mean if the determinant of a matrix is zero? 5. How can we check if a system of equations is consistent? **Tip:** When solving systems of equations, always check if the matrix is invertible to determine if there is a unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
System of Equations
Formulas
Inverse of a matrix A: A^{-1}
Determinant of a matrix
Conditions for a unique solution in a system of linear equations
Theorems
A system of linear equations has a unique solution if and only if the coefficient matrix is invertible.
A matrix is invertible if its determinant is non-zero.
Suitable Grade Level
Undergraduate
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