Question

Solve by matrix method the system of equations:

\[ \begin{aligned} x - y + z &= 4 \\ 2x + y - 3z &= 0 \\ x + y + z &= 2 \end{aligned} \]

Hint:

When solving systems of linear equations using matrices, represent the system in matrix form, calculate the inverse of the coefficient matrix, and multiply by the constant vector to get the solution.

Answer 1

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 \( x = 3, y = -1, z = 2 \).