Question

If \( \begin{pmatrix} x+y & x-y \\ 2x+z & x+z \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} \), then the values of \( x, y \) and \( z \) are respectively:

• A. \( 0, 0, 1 \)
• B. \( 1, 1, 0 \)
• C. \( -1, 0, 0 \)
• D. \( 0, 0, 0 \)
• E. \( 1, 1, 1 \)

Hint:

When solving matrix equality, look for the simplest equations first. Here, \( x+y=0 \) and \( x-y=0 \) immediately imply both variables must be zero.

Answer 1

The Correct Option is A

Concept: Two matrices are equal if and only if their corresponding elements are equal. This leads to a system of linear equations. Solving these equations allows us to determine the unique values of the variables \( x, y, \) and \( z \).

Step 1: Equating corresponding elements to form equations.
From the first row, we get: \[ 1) \quad x + y = 0 \] \[ 2) \quad x - y = 0 \] From the second row, we get: \[ 3) \quad 2x + z = 1 \] \[ 4) \quad x + z = 1 \]

Step 2: Solving the system of equations.
Adding equations (1) and (2): \[ (x + y) + (x - y) = 0 + 0 \implies 2x = 0 \implies x = 0 \] Substituting \( x = 0 \) into equation (1): \[ 0 + y = 0 \implies y = 0 \] Substituting \( x = 0 \) into equation (4): \[ 0 + z = 1 \implies z = 1 \] Thus, the values are \( x = 0, y = 0, z = 1 \).