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Решить обратной матрицой

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

The system is:
\[
\begin{cases}
x + y - z = 1 \\
8x + 3y - 6z = 2 \\
4x + y - 3z = 3
\end{cases}
\]

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.

Learn to solve a system of equations using matrix inversion. The system x + y - z = 1, 8x + 3y - 6z = 2, and 4x + y - 3z = 3 is solved by expressing it in matrix form and applying the inverse matrix method. This detailed solution covers matrix inversion and linear algebra concepts, suitable for high school students.

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