If A is an invertible matrix of order 3 and B is another matrix of the same order as of A, such that `|B|=2, A^(T)|A|B=A|B|B^(T).` If `|AB^(-1)adj(A^(T)B)^(-1)|=K`, then the value of 4K is equal to

Answer

To solve the problem step by step, we will analyze the given information and apply the properties of determinants and matrices accordingly.

### Given:
- \( A \) is an invertible matrix of order 3.
- \( B \) is another matrix of the same order as \( A \) such that \( |B| = 2 \).
- The equation \( A^T |A| B = A |B| B^T \) holds.
- We need to find \( |AB^{-1} \text{adj}(A^T B)^{-1}| = K \) and then find the value of \( 4K \).

### Step 1: Analyze the given equation
We start with the equation:
\[ A^T |A| B = A |B| B^T \]
Taking the determinant on both sides:
\[ |A^T| |A| |B| = |A| |B| |B^T| \]

### Step 2: Simplify using properties of determinants
Using the property \( |A^T| = |A| \) and \( |B^T| = |B| \):
\[ |A|^2 |B| = |A| |B|^2 \]
Dividing both sides by \( |A| |B| \) (since both are non-zero):
\[ |A| = |B| \]

### Step 3: Substitute known values
Given \( |B| = 2 \), we find:
\[ |A| = 2 \]

### Step 4: Find \( |AB^{-1} \text{adj}(A^T B)^{-1}| \)
We know:
\[ |AB^{-1} \text{adj}(A^T B)^{-1}| = |A| |B^{-1}| |\text{adj}(A^T B)|^{-1} \]
Using the property \( |B^{-1}| = \frac{1}{|B|} \):
\[ |B^{-1}| = \frac{1}{2} \]

### Step 5: Calculate \( |\text{adj}(A^T B)| \)
Using the property of adjugate matrices:
\[ |\text{adj}(A^T B)| = |A^T B|^{n-1} = |A^T|^{2} |B|^{2} = |A|^{2} |B|^{2} \]
Substituting the known values:
\[ |\text{adj}(A^T B)| = (2^2)(2^2) = 4 \cdot 4 = 16 \]

### Step 6: Substitute values back into the expression
Now we can substitute back into our expression:
\[ |AB^{-1} \text{adj}(A^T B)^{-1}| = |A| \cdot |B^{-1}| \cdot |\text{adj}(A^T B)|^{-1} \]
Substituting the values:
\[ = 2 \cdot \frac{1}{2} \cdot \frac{1}{16} = \frac{2}{2 \cdot 16} = \frac{1}{16} \]

### Step 7: Find \( K \) and \( 4K \)
Thus, we have:
\[ K = \frac{1}{16} \]
Now, calculating \( 4K \):
\[ 4K = 4 \cdot \frac{1}{16} = \frac{4}{16} = \frac{1}{4} \]

### Final Answer
The value of \( 4K \) is \( \frac{1}{4} \).