Question

If the matrix $A = \begin{bmatrix} 0 & x + y & 1 \\ 3 & z & 2 \\ x - y & -2 & 0 \end{bmatrix}$ is skew-symmetric, then:

• A. \(x = 2, y = 1, z = 0\)
• B. \(x = 2, y = 2, z = 0\)
• C. \(x = -2, y = -1, z = 0\)
• D. \(x = -2, y = -1, z = -1\)

Hint:

In a skew-symmetric matrix, diagonal elements are zero and \(a_{ij} = -a_{ji}\) helps to find variables systematically.

Answer 1

The Correct Option is C

For a matrix \(A\) to be skew-symmetric, it must satisfy: \[ A^T = -A \] This means \(a_{ij} = -a_{ji}\) and the diagonal elements must be zero. Given matrix: \[ A = \begin{bmatrix} 0 & x + y & 1 \\ 3 & z & 2 \\ x - y & -2 & 0 \end{bmatrix} \] Step 1: Diagonal elements must be zero: \[ a_{11} = 0, \quad a_{22} = z, \quad a_{33} = 0 \] So, \[ z = 0 \] Step 2: Off-diagonal elements satisfy: \[ a_{ij} = -a_{ji} \] From element (1,2) and (2,1): \[ x + y = -3 \] From element (1,3) and (3,1): \[ 1 = -(x - y) \implies 1 = -x + y \implies x - y = -1 \] From element (2,3) and (3,2): \[ 2 = -(-2) = 2 \] (This is consistent.) Step 3: Solve the system of equations: \[ x + y = -3 \quad (1) \] \[ x - y = -1 \quad (2) \] Add (1) and (2): \[ (x + y) + (x - y) = -3 + (-1) \implies 2x = -4 \implies x = -2 \] Substitute \(x = -2\) into (1): \[ -2 + y = -3 \implies y = -1 \] Thus, \[ x = -2, \quad y = -1, \quad z = 0 \]