Math Problem Statement
solve this inverse matrix
Solution
To solve this system of linear equations using the inverse matrix method, we represent the system in matrix form , where is the coefficient matrix, is the variable matrix, and is the constants matrix.
The given system is:
x - 2y = 3 \\ 2x + y + 2z = 5 \\ 3x - y - z = 2 \end{cases}$$ ### Step 1: Write the System in Matrix Form The matrix form $$AX = B$$ is: $$\begin{bmatrix} 1 & -2 & 0 \\ 2 & 1 & 2 \\ 3 & -1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3 \\ 5 \\ 2 \end{bmatrix}$$ ### Step 2: Find the Inverse of Matrix $$A$$ If $$A^{-1}$$ exists, then the solution can be found using: $$X = A^{-1} B$$ I’ll calculate this and provide the solution for $$x$$, $$y$$, and $$z$$.The solution to the system using the inverse matrix method is: $$x = 1, \quad y = -1, \quad z = 2$$ This matches the provided solution. Would you like further details on the steps taken, or do you have any questions? Here are some related questions to consider: 1. How do you determine if a matrix is invertible? 2. What are the implications if the matrix $$A$$ in $$AX = B$$ has no inverse? 3. Can the inverse matrix method be applied to systems with more than three variables? 4. How does one compute the inverse of a 3x3 matrix by hand? 5. How does the solution method change if there is no unique solution? **Tip:** The inverse matrix method is efficient for small systems, but for larger systems, other techniques (like Gaussian elimination) might be more practical.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 AX = B
Inverse matrix solution formula: X = A^(-1)B
Theorems
Inverse Matrix Theorem
Properties of Matrix Multiplication
Suitable Grade Level
College Level
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