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 start with the information given in the question.

### Step 1: Understanding the Determinants
We are given that the determinants of the matrices \( A \), \( 4A^{-1} \), and \( A^T \) are equal. Therefore, we can set up the following equation:

\[ |A| = |4A^{-1}| \]

### Step 2: Using the Properties of Determinants
We know that the determinant of a scalar multiple of a matrix can be expressed as:

\[ |kA| = k^n |A| \]

where \( n \) is the order of the matrix. In this case, since \( A \) is a \( 3 \times 3 \) matrix, \( n = 3 \). Thus, we have:

\[ |4A^{-1}| = 4^3 |A^{-1}| = 64 |A^{-1}| \]

### Step 3: Relating Determinants
Since \( |A^{-1}| = \frac{1}{|A|} \), we can substitute this into our equation:

\[ |A| = 64 \cdot \frac{1}{|A|} \]

### Step 4: Rearranging the Equation
Multiplying both sides by \( |A| \) gives us:

\[ |A|^2 = 64 \]

Taking the square root of both sides, we find:

\[ |A| = 8 \quad \text{or} \quad |A| = -8 \]

### Step 5: Using the Adjoint Condition
We are also given that \( \text{adj}(A) = 2A^T \). The relationship between the determinant and the adjoint is given by:

\[ \text{adj}(A) = |A| A^{-1} \]

Thus, we can equate:

\[ |A| A^{-1} = 2A^T \]

Taking the determinant on both sides gives:

\[ |\text{adj}(A)| = |2A^T| \]

### Step 6: Determinant of the Adjoint
The determinant of the adjoint of a matrix is given by:

\[ |\text{adj}(A)| = |A|^{n-1} = |A|^2 \quad \text{(since } n = 3\text{)} \]

Thus, we have:

\[ |A|^2 = |2A^T| \]

### Step 7: Determinant of \( 2A^T \)
Using the properties of determinants, we can express \( |2A^T| \):

\[ |2A^T| = 2^3 |A| = 8 |A| \]

### Step 8: Setting the Equations Equal
Now we can set the two expressions for \( |\text{adj}(A)| \) equal to each other:

\[ |A|^2 = 8 |A| \]

### Step 9: Solving for \( |A| \)
Rearranging gives us:

\[ |A|^2 - 8|A| = 0 \]

Factoring out \( |A| \):

\[ |A|(|A| - 8) = 0 \]

Thus, \( |A| = 0 \) or \( |A| = 8 \). Since \( |A| \) cannot be zero (as it would imply \( A \) is singular), we have:

\[ |A| = 8 \]

### Step 10: Finding \( |3A^{-1}| \)
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:

\[ \boxed{\frac{27}{8}} \]