If A is a skew-symmetric matrix such that `(A^(n))'=kA^(n), n in N`, then the value of k is

Options

A. 1
B. `-1`
C. `(-1)^(n)`
D. n

Answer

To solve the problem, we need to find the value of \( k \) in the equation \( (A^n)' = kA^n \) where \( A \) is a skew-symmetric matrix.

### Step-by-step Solution:

1. **Understand Skew-Symmetric Matrix**:
A matrix \( A \) is skew-symmetric if \( A' = -A \).

2. **Apply the Property of Transpose**:
For any matrix \( A \), the transpose of the product of matrices follows the rule:
\[ (AB)' = B'A' \]
For powers of a matrix, we have:
\[ (A^n)' = (A \cdot A \cdots A)' = A' \cdot A' \cdots A' = (A')^n \]

3. **Substituting the Skew-Symmetric Property**:
Since \( A' = -A \), we can substitute this into our equation:
\[ (A^n)' = (-A)^n = (-1)^n A^n \]

4. **Set Up the Given Equation**:
The problem states that:
\[ (A^n)' = k A^n \]
From our previous step, we have:
\[ (A^n)' = (-1)^n A^n \]

5. **Equate the Two Expressions**:
Now, we can equate the two expressions we have for \( (A^n)' \):
\[ k A^n = (-1)^n A^n \]

6. **Solve for \( k \)**:
Since \( A^n \) is not the zero matrix (as \( A \) is skew-symmetric and not the zero matrix), we can divide both sides by \( A^n \):
\[ k = (-1)^n \]

### Conclusion:
Thus, the value of \( k \) is:
\[ k = (-1)^n \]