Given the matrix equation \( AX = B \), we have the following system of linear equations: \[ x - y + z = 4 \quad \text{(Equation 1)} \] \[ 2x + y - 3z = 0 \quad \text{(Equation 2)} \] \[ x + y + z = 2 \quad \text{(Equation 3)}. \]
Step 1: Express \( y \) in terms of \( x \) and \( z \) From Equation (3), isolate \( y \): \[ x + y + z = 2 \quad \Rightarrow \quad y = 2 - x - z \quad \text{(Equation 4)}. \]
Step 2: Substitute Equation (4) into Equation (1) Substitute the expression for \( y \) into Equation (1): \[ x - (2 - x - z) + z = 4 \] Simplify the equation: \[ x - 2 + x + z + z = 4 \quad \Rightarrow \quad 2x + 2z - 2 = 4 \quad \Rightarrow \quad 2x + 2z = 6 \quad \Rightarrow \quad x + z = 3 \quad \text{(Equation 5)}. \]
Step 3: Substitute Equation (4) into Equation (2) Substitute the expression for \( y \) into Equation (2): \[ 2x + (2 - x - z) - 3z = 0 \] Simplify the equation: \[ 2x + 2 - x - z - 3z = 0 \quad \Rightarrow \quad x - 4z + 2 = 0 \quad \Rightarrow \quad x = 4z - 2 \quad \text{(Equation 6)}. \]
Step 4: Substitute Equation (6) into Equation (5) Substitute the expression for \( x \) into Equation (5): \[ (4z - 2) + z = 3 \] Simplify the equation: \[ 4z - 2 + z = 3 \quad \Rightarrow \quad 5z = 5 \quad \Rightarrow \quad z = 1. \]
Step 5: Determine the values of \( x \) and \( y \) Substitute \( z = 1 \) into Equation (6) to find \( x \): \[ x = 4(1) - 2 = 4 - 2 = 2. \] Substitute \( x = 2 \) and \( z = 1 \) into Equation (4) to find \( y \): \[ y = 2 - 2 - 1 = -1. \] Evaluate the expression \( 2x + y - z \) using the obtained values: \[ 2x + y - z = 2(2) + (-1) - 1 = 4 - 1 - 1 = 2. \] Thus, \( 2x + y - z = 2\).
Conclusion. The value of \( 2x + y - z \) is 2.