Let `A = [a_(ij)] " be a " 3 xx3` matrix and let `A_(1)` denote the matrix of the cofactors of elements of matrix A and `A_(2)` be the matrix of cofactors of elements of matrix `A_(1)` and so on. If `A_(n)` denote the matrix of cofactros of elements of matrix `A_(n -1)`, then `|A_(n)|` equals

Options

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

Answer

To solve the problem, we need to analyze the relationship between the determinants of the matrix \( A \) and its cofactors. Let's denote the determinant of matrix \( A \) as \( |A| \).

### Step-by-Step Solution:

1. **Understanding the Cofactor Matrix**:
The cofactor matrix \( A_1 \) of a matrix \( A \) is defined such that each element \( a_{ij} \) of \( A \) is replaced by its cofactor \( C_{ij} \). The determinant of the cofactor matrix \( A_1 \) can be expressed in terms of the determinant of \( A \):
\[ |A_1| = |A|^2 \]

2. **Finding the Determinant of \( A_2 \)**:
Now, we need to find the determinant of the cofactor matrix \( A_2 \) of \( A_1 \):
\[ |A_2| = |A_1|^2 = (|A|^2)^2 = |A|^4 \]

3. **Continuing the Pattern**:
We can continue this process. The determinant of \( A_3 \), which is the cofactor matrix of \( A_2 \), can be calculated as:
\[ |A_3| = |A_2|^2 = (|A|^4)^2 = |A|^8 \]

4. **Generalizing the Pattern**:
From the above steps, we can see a pattern emerging. For each \( n \):
\[ |A_n| = |A_{n-1}|^2 \]
This leads us to the conclusion that:
\[ |A_n| = |A|^{2^n} \]

5. **Final Result**:
Therefore, the determinant of the matrix of cofactors \( A_n \) is given by:
\[ |A_n| = |A|^{2^n} \]

### Conclusion:
The final answer is:
\[ |A_n| = |A|^{2^n} \]