If A is a square matric of order 5 and `2A^(-1)=A^(T)`, then the remainder when `|"adj. (adj. (adj. A))"|` is divided by 7 is

Options

A. 2
B. 3
C. 4
D. 5

Explanation

To solve the problem, we need to find the remainder when \(|\text{adj}(\text{adj}(\text{adj}(A)))|\) is divided by 7, given that \(2A^{-1} = A^T\) for a square matrix \(A\) of order 5.

### Step-by-step Solution:

1. **Start with the given equation:**
\[ 2A^{-1} = A^T \]
Rearranging gives:
\[ A^T = 2A^{-1} \]

2. **Take the determinant of both sides:**
\[ |A^T| = |2A^{-1}| \]
Using properties of determinants, we have:
\[ |A^T| = |A| \quad \text{and} \quad |2A^{-1}| = 2^5 |A|^{-1} = 32 |A|^{-1} \]
Thus, we can write:
\[ |A| = 32 |A|^{-1} \]

3. **Multiply both sides by \(|A|\):**
\[ |A|^2 = 32 \]
Therefore:
\[ |A| = \sqrt{32} = 4\sqrt{2} \]

4. **Find the value of \(|\text{adj}(A)|\):**
The determinant of the adjugate of a matrix \(A\) of order \(n\) is given by:
\[ |\text{adj}(A)| = |A|^{n-1} \]
For our case, \(n = 5\):
\[ |\text{adj}(A)| = |A|^{4} = (4\sqrt{2})^{4} = 256 \cdot 4 = 1024 \]

5. **Find \(|\text{adj}(\text{adj}(A))|\):**
Using the same property:
\[ |\text{adj}(\text{adj}(A))| = |\text{adj}(A)|^{4} = (1024)^{4} = 1024^{4} \]

6. **Find \(|\text{adj}(\text{adj}(\text{adj}(A)))|\):**
Again applying the property:
\[ |\text{adj}(\text{adj}(\text{adj}(A)))| = |\text{adj}(\text{adj}(A))|^{4} = (1024^{4})^{4} = 1024^{16} \]

7. **Calculate \(1024 \mod 7\):**
First, find \(1024 \mod 7\):
\[ 1024 \div 7 = 146 \quad \text{remainder } 2 \]
Thus:
\[ 1024 \equiv 2 \mod 7 \]

8. **Now calculate \(1024^{16} \mod 7\):**
Since \(1024 \equiv 2 \mod 7\):
\[ 1024^{16} \equiv 2^{16} \mod 7 \]

9. **Use Fermat's Little Theorem:**
Since 7 is prime:
\[ 2^{6} \equiv 1 \mod 7 \]
Therefore:
\[ 2^{16} = 2^{6 \cdot 2 + 4} \equiv (2^6)^2 \cdot 2^4 \equiv 1^2 \cdot 2^4 \equiv 2^4 \mod 7 \]

10. **Calculate \(2^4 \mod 7\):**
\[ 2^4 = 16 \equiv 2 \mod 7 \]

11. **Final result:**
The remainder when \(|\text{adj}(\text{adj}(\text{adj}(A)))|\) is divided by 7 is:
\[ \boxed{2} \]