When dealing with skew-symmetric matrices, remember that the elements satisfy the property \( a_{ij} = -a_{ji} \), which means each element is the negative of its corresponding off-diagonal element. Diagonal elements are always zero. Use this property to solve for unknowns and verify the consistency of the matrix. In such problems, carefully apply the skew-symmetric condition to each pair of elements and solve the resulting equations.
The provided matrix is: \( \begin{bmatrix} 0 & -1 & 3x \\ 1 & y & -5 \\ -6 & 5 & 0 \end{bmatrix} \). A matrix \(A\) is skew-symmetric if its transpose \(A^T\) equals its negative \(-A\). This implies that for any element \(a_{ij}\), the condition \(a_{ij} = -a_{ji}\) must hold.
Comparing elements at positions \((1,2)\) and \((2,1)\): \( -1 = -1 \). This condition is met.
Comparing elements at positions \((1,3)\) and \((3,1)\): \( 3x = 6 \), which simplifies to \( x = 2 \).
Comparing elements at positions \((2,3)\) and \((3,2)\): \( -5 = -y \), which simplifies to \( y = 5 \).
Now, we calculate \( 5x - y \): \( 5(2) - 5 = 10 - 5 = 10 \).
Therefore, the value of \( 5x - y \) is 10.