Math Problem Statement

Решить обратной матрицой

Решить обратной матрицой

Solution

To solve this system of equations using the inverse matrix method, we start by expressing it in matrix form:

The system is:

x + y - z = 1 \\ 8x + 3y - 6z = 2 \\ 4x + y - 3z = 3 \end{cases}$$ This can be rewritten in matrix form as $$AX = B$$, where: $$A = \begin{pmatrix} 1 & 1 & -1 \\ 8 & 3 & -6 \\ 4 & 1 & -3 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$$ ### Steps: 1. **Find the inverse of matrix $$A$$**, if it exists. 2. Multiply both sides by $$A^{-1}$$ to isolate $$X$$: $$X = A^{-1} B$$. I'll calculate $$A^{-1}$$ and then solve for $$X$$.The solution to the system of equations is: $$x = -8, \quad y = -4, \quad z = -13$$ Would you like a detailed explanation of the steps, or do you have any further questions? ### Related Questions: 1. How do we know if a matrix is invertible? 2. What happens if the matrix $$A$$ is not invertible? 3. Can the inverse matrix method be applied to larger systems? 4. How does this method compare to Gaussian elimination? 5. How do rounding errors affect the accuracy of matrix inversions in computational solutions? **Tip:** Inverting a matrix is computationally intensive for large systems. For efficiency, consider other methods like LU decomposition for larger matrices.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Systems of Linear Equations

Formulas

AX = B
X = A^-1 * B

Theorems

Inverse Matrix Theorem

Suitable Grade Level

Grades 11-12