Solve by matrix method the system of equations:
\[
\begin{aligned}
x - y + z &= 4 \\
2x + y - 3z &= 0 \\
x + y + z &= 2
\end{aligned}
\]
Solve by matrix method the system of equations:
\[ \begin{aligned} x - y + z &= 4 \\ 2x + y - 3z &= 0 \\ x + y + z &= 2 \end{aligned} \]
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Solution and Explanation
Step 1: Write the system of equations in matrix form:
\[ \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix} \]
Step 2: Find the inverse of the coefficient matrix:
\[ A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix} \]
Step 3: Use the formula \( \vec{x} = A^{-1} \cdot \vec{b} \) to solve for \( \vec{x} \), where \( \vec{x} \) represents the vector of variables and \( \vec{b} \) is the constant vector.
After calculating \( A^{-1} \) and multiplying by \( \vec{b} \), the solution is:
\[ x = 3, \quad y = -1, \quad z = 2 \]
Thus, the solution to the system of equations is \( x = 3, y = -1, z = 2 \).
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