Question

If \( AX = B \), where \[ A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}, \] then \( 2x + y - z \) is:

• A. 2
• B. 1
• C. 4
• D. -2

Hint:

To solve a system of linear equations, use substitution or elimination methods to simplify and solve for the unknowns.

Answer 1

The Correct Option is A

We are given the matrix equation \( AX = B \), which expands to the following system of 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: Solve for \( y \) in terms of \( x \) and \( z \) From Equation (3), solve for \( y \): \[ x + y + z = 2 \quad \Rightarrow \quad y = 2 - x - z \quad \text{(Equation 4)}. \] 

Step 2: Substitute \( y = 2 - x - z \) into Equation (1) Substitute Equation (4) into Equation (1): \[ x - (2 - x - z) + z = 4 \] Simplifying: \[ x - 2 + x + z + z = 4 \quad \Rightarrow \quad 2x + 2z - 2 = 4 \quad \Rightarrow \quad 2x + 2z = 6 \quad \Rightarrow \quad x + z = 3 \quad \text{(Equation 5)}. \] 

Step 3: Substitute \( y = 2 - x - z \) into Equation (2) Substitute Equation (4) into Equation (2): \[ 2x + (2 - x - z) - 3z = 0 \] Simplifying: \[ 2x + 2 - x - z - 3z = 0 \quad \Rightarrow \quad x - 4z + 2 = 0 \quad \Rightarrow \quad x = 4z - 2 \quad \text{(Equation 6)}. \] 

Step 4: Substitute \( x = 4z - 2 \) into Equation (5) Substitute Equation (6) into Equation (5): \[ (4z - 2) + z = 3 \] Simplifying: \[ 4z - 2 + z = 3 \quad \Rightarrow \quad 5z = 5 \quad \Rightarrow \quad z = 1. \] 

Step 5: Solve for \( x \) 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 \( y \): \[ y = 2 - 2 - 1 = -1. \] Substitute these values into the expression \( 2x + y - z \): \[ 2x + y - z = 2 * 2 + (-1) - 1 = 4 - 1 - 1 = 2 \] Thus, \( 2x + y - z = 2\). 

Conclusion. Therefore, \( 2x + y - z = 2 \).