A is a square matrix of order `3xx3`. The matrices `A, 4A^(-1), A^(T)` all have the same value of the determinant. If `(adjA)=2A^(T).` then `|3A^(-1)|` is equal to

Options

A. `(3)/(2)`
B. `(9)/(4)`
C. `(27)/(8)`
D. 3

Answer

To solve the problem step by step, we need to analyze the given conditions and use properties of determinants.

### Step 1: Understand the given conditions
We are given that:
1. \( |A| = |4A^{-1}| = |A^T| \)
2. \( \text{adj}(A) = 2A^T \)

### Step 2: Use the property of determinants
Using the property of determinants, we know:
\[ |kA| = k^n |A| \]
where \( n \) is the order of the matrix. Since \( A \) is a \( 3 \times 3 \) matrix, we have:
\[ |4A^{-1}| = 4^3 |A^{-1}| = 64 |A^{-1}| \]
Also, we know that \( |A^{-1}| = \frac{1}{|A|} \). Thus:
\[ |4A^{-1}| = 64 \cdot \frac{1}{|A|} = \frac{64}{|A|} \]

### Step 3: Set up the equation
From the first condition, we have:
\[ |A| = \frac{64}{|A|} \]
Multiplying both sides by \( |A| \):
\[ |A|^2 = 64 \]
Taking the square root:
\[ |A| = \pm 8 \]

### Step 4: Use the adjoint condition
Next, we analyze the condition \( \text{adj}(A) = 2A^T \). The determinant of the adjoint of a matrix is given by:
\[ |\text{adj}(A)| = |A|^{n-1} = |A|^2 \quad \text{(since \( n = 3 \))} \]
Thus:
\[ |\text{adj}(A)| = |A|^2 \]
Also, using the property of determinants:
\[ |\text{adj}(A)| = |2A^T| = 2^3 |A| = 8 |A| \]
Equating the two expressions for \( |\text{adj}(A)| \):
\[ |A|^2 = 8 |A| \]
Rearranging gives:
\[ |A|^2 - 8 |A| = 0 \]
Factoring out \( |A| \):
\[ |A| (|A| - 8) = 0 \]
This implies:
\[ |A| = 0 \quad \text{or} \quad |A| = 8 \]
Since we already found \( |A| = \pm 8 \), we conclude:
\[ |A| = 8 \]

### Step 5: Calculate \( |3A^{-1}| \)
Now, we need to find \( |3A^{-1}| \):
\[ |3A^{-1}| = 3^3 |A^{-1}| = 27 \cdot \frac{1}{|A|} = 27 \cdot \frac{1}{8} = \frac{27}{8} \]

### Final Answer
Thus, the value of \( |3A^{-1}| \) is:
\[ \boxed{\frac{27}{8}} \]