If `[[x,y^(3)],[2,0]]= [[1,8],[2,0]]`, then find the corresponding skew - symmetric matric of `[[x,y],[2,0]]`.

Answer

To solve the problem, we need to find the corresponding skew-symmetric matrix of the matrix \(\begin{bmatrix} x & y \\ 2 & 0 \end{bmatrix}\) given that \(\begin{bmatrix} x & y^3 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 8 \\ 2 & 0 \end{bmatrix}\).

### Step-by-Step Solution:

1. **Set Up the Equation**:
We have the equation:
\[ \begin{bmatrix} x & y^3 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 8 \\ 2 & 0 \end{bmatrix} \]
From this, we can equate the corresponding elements of the matrices.

2. **Equate the Elements**:
From the first row and first column:
\[ x = 1 \]
From the first row and second column:
\[ y^3 = 8 \]
From the second row and first column:
\[ 2 = 2 \quad \text{(This is always true)} \]
From the second row and second column:
\[ 0 = 0 \quad \text{(This is always true)} \]

3. **Solve for \(y\)**:
To find \(y\), we solve the equation \(y^3 = 8\):
\[ y = \sqrt[3]{8} = 2 \]

4. **Construct the Matrix**:
Now that we have \(x\) and \(y\):
\[ x = 1, \quad y = 2 \]
The matrix \(\begin{bmatrix} x & y \\ 2 & 0 \end{bmatrix}\) becomes:
\[ A = \begin{bmatrix} 1 & 2 \\ 2 & 0 \end{bmatrix} \]

5. **Find the Skew-Symmetric Matrix**:
The skew-symmetric matrix \(B\) corresponding to matrix \(A\) is given by:
\[ B = \frac{1}{2}(A - A^T) \]
First, we calculate \(A^T\):
\[ A^T = \begin{bmatrix} 1 & 2 \\ 2 & 0 \end{bmatrix}^T = \begin{bmatrix} 1 & 2 \\ 2 & 0 \end{bmatrix} \]
Now, we compute \(A - A^T\):
\[ A - A^T = \begin{bmatrix} 1 & 2 \\ 2 & 0 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

6. **Final Skew-Symmetric Matrix**:
Therefore, the skew-symmetric matrix \(B\) is:
\[ B = \frac{1}{2} \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

### Conclusion:
The corresponding skew-symmetric matrix of \(\begin{bmatrix} x & y \\ 2 & 0 \end{bmatrix}\) is:
\[ \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]