If A is a square matrix such that `|A| = 2`, then for any positive integer `n, |A^(n)|` is equal to

Options

A. `2^(n)`
B. `n^(2)`
C. 0
D. 2n

Explanation

To solve the problem, we need to find the determinant of \( A^n \) given that the determinant of matrix \( A \) is \( |A| = 2 \).

### Step-by-Step Solution:

1. **Understand the property of determinants**:
The determinant of the product of two matrices is equal to the product of their determinants. That is, for any two square matrices \( B \) and \( C \), we have:
\[ |BC| = |B| \cdot |C| \]

2. **Apply the property to powers of a matrix**:
For any positive integer \( n \), we can express \( A^n \) as the product of \( A \) multiplied by itself \( n \) times:
\[ A^n = A \cdot A \cdot A \cdots A \quad (n \text{ times}) \]

3. **Use the determinant property**:
Applying the determinant property to \( A^n \):
\[ |A^n| = |A \cdot A \cdot A \cdots A| = |A| \cdot |A| \cdots |A| \quad (n \text{ times}) = |A|^n \]

4. **Substitute the given value of \( |A| \)**:
We know from the problem that \( |A| = 2 \). Therefore, we can substitute this value into the equation:
\[ |A^n| = |A|^n = 2^n \]

5. **Conclusion**:
Thus, for any positive integer \( n \), the determinant of \( A^n \) is:
\[ |A^n| = 2^n \]

### Final Answer:
\[ |A^n| = 2^n \]