If the matrix `[{:(0,-1,3x),(1,y,-5),(-6,5,0):}]` is skew- symmetric, then

Options

A. `x=-2,y=0`
B. `x=2,y=0`
C. `x=-2,y=1`
D. `x=2,y=-2`

Answer

To determine the values of \( x \) and \( y \) such that the matrix

\[ A = \begin{pmatrix} 0 & -1 & 3x \\ 1 & y & -5 \\ -6 & 5 & 0 \end{pmatrix} \]

is skew-symmetric, we follow these steps:

### Step 1: Understand the property of skew-symmetric matrices
A matrix \( A \) is skew-symmetric if \( A^T = -A \), where \( A^T \) is the transpose of \( A \).

### Step 2: Find the transpose of the matrix \( A \)
The transpose of matrix \( A \) is obtained by swapping rows with columns:

\[ A^T = \begin{pmatrix} 0 & 1 & -6 \\ -1 & y & 5 \\ 3x & -5 & 0 \end{pmatrix} \]

### Step 3: Set up the equation \( A^T = -A \)
Now we set \( A^T \) equal to \( -A \):

\[ \begin{pmatrix} 0 & 1 & -6 \\ -1 & y & 5 \\ 3x & -5 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 & -3x \\ 1 & -y & 5 \\ 6 & -5 & 0 \end{pmatrix} \]

### Step 4: Equate corresponding elements
From the equation \( A^T = -A \), we can equate the corresponding elements:

1. From the (1,2) entry: \( 1 = -1 \) (which is not useful)
2. From the (2,1) entry: \( -1 = 1 \) (which is also not useful)
3. From the (1,3) entry: \( -6 = -3x \)
4. From the (2,2) entry: \( y = -y \)
5. From the (3,1) entry: \( 3x = 6 \)

### Step 5: Solve for \( y \)
From the equation \( y = -y \):

\[ y + y = 0 \implies 2y = 0 \implies y = 0 \]

### Step 6: Solve for \( x \)
From the equation \( -6 = -3x \):

\[ -6 = -3x \implies 6 = 3x \implies x = 2 \]

### Final Result
Thus, the values of \( x \) and \( y \) are:

\[ x = 2, \quad y = 0 \]