A matrix \(A\) is skew-symmetric if \(A^T = -A\). This condition implies that \(a_{ij} = -a_{ji}\) and all 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: Ensure diagonal elements are zero.
The diagonal elements are \(a_{11} = 0\), \(a_{22} = z\), and \(a_{33} = 0\). For skew-symmetry, \(z\) must be zero.
\[z = 0\]
Step 2: Verify off-diagonal elements satisfy \(a_{ij} = -a_{ji}\).
- For elements (1,2) and (2,1): \(a_{12} = x + y\) and \(a_{21} = 3\). Thus, \(x + y = -3\).
- For elements (1,3) and (3,1): \(a_{13} = 1\) and \(a_{31} = x - y\). Thus, \(1 = -(x - y)\), which simplifies to \(x - y = -1\).
- For elements (2,3) and (3,2): \(a_{23} = 2\) and \(a_{32} = -2\). Thus, \(2 = -(-2)\), which is \(2 = 2\). This pair is consistent.
Step 3: Solve the system of equations for \(x\) and \(y\).
We have the system:
1. \(x + y = -3\)
2. \(x - y = -1\)
Adding equations (1) and (2):
\[(x + y) + (x - y) = -3 + (-1) \implies 2x = -4 \implies x = -2\]
Substitute \(x = -2\) into equation (1):
\[-2 + y = -3 \implies y = -1\]
The values are \(x = -2\), \(y = -1\), and \(z = 0\).