Question

If the matrix \(\begin {bmatrix} x-y&1&-2 \\ 2x-y&0&3 \\ 2&-3&0 \end {bmatrix}\) is skew-symmetric, then values of x and y are respectively

• A. \(\frac{1}{2},1\)
• B. \(1,\frac{1}{2}\)
• C. 1,1
• D. -1,-1

Answer 1

The Correct Option is D

A matrix is skew-symmetric if for a square matrix \( A \), \( A^T = -A \), where \( A^T \) is the transpose of \( A \). For the matrix \(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\), skew-symmetry conditions are: \( a = 0 \), \( e = 0 \), \( i = 0 \), \( b = -d \), \( c = -g \), \( f = -h \).

Given matrix:\(\begin{bmatrix} x-y & 1 & -2 \\ 2x-y & 0 & 3 \\ 2 & -3 & 0 \end{bmatrix}\).

Apply skew-symmetric conditions:

1. \(x-y = 0\), because diagonal element must be zero.
2. \(2x-y = -1\) (as \(1 = -(2x-y)\)).
3. \(-2 = -2\), satisfied.
4. \(3 = 3\), satisfied.

From \(x-y = 0\), we have:

\(x = y\)

Substituting in \(2x-y = -1\):

\(2x-x = -1\)

\(x = -1\)

Since \(x = y\), \(y = -1\).

The values of \(x\) and \(y\) are \(-1, -1\).