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 properties of determinants and matrices.

### Step 1: Understand the given equation
We are given the equation:
\[ A^{T} |A| B = A |B| B^{T} \]
where \( |B| = 2 \).

### Step 2: Take determinants on both sides
Taking the determinant of both sides, we have:
\[ |A^{T}| |A| |B| = |A| |B| |B^{T}| \]

### Step 3: Use properties of determinants
Using the property that \( |A^{T}| = |A| \) and \( |B^{T}| = |B| \), we can simplify the equation:
\[ |A|^2 |B| = |A| |B|^2 \]

### Step 4: Cancel out common terms
Since \( |B| = 2 \) and \( |A| \neq 0 \) (because A is invertible), we can divide both sides by \( |A| |B| \):
\[ |A| = |B| \]

### Step 5: Substitute the value of |B|
Substituting \( |B| = 2 \):
\[ |A| = 2 \]

### Step 6: Find the expression for \( |AB^{-1} \text{adj}(A^{T}B)^{-1}| \)
We need to find:
\[ |AB^{-1} \text{adj}(A^{T}B)^{-1}| \]

### Step 7: Use determinant properties
Using the property of determinants:
\[ |AB^{-1} \text{adj}(A^{T}B)^{-1}| = |A| |B^{-1}| |\text{adj}(A^{T}B)| \]

### Step 8: Calculate each component
1. **Calculate \( |B^{-1}| \)**:
\[ |B^{-1}| = \frac{1}{|B|} = \frac{1}{2} \]

2. **Calculate \( |\text{adj}(A^{T}B)| \)**:
Using the property \( |\text{adj}(X)| = |X|^{n-1} \) for a matrix of order \( n \):
\[ |\text{adj}(A^{T}B)| = |A^{T}B|^{2} \]
Since \( |A^{T}B| = |A| |B| = 2 \times 2 = 4 \):
\[ |\text{adj}(A^{T}B)| = 4^{2} = 16 \]

### Step 9: Combine the results
Now substituting back into our expression:
\[ |AB^{-1} \text{adj}(A^{T}B)^{-1}| = |A| \cdot |B^{-1}| \cdot |\text{adj}(A^{T}B)| \]
\[ = 2 \cdot \frac{1}{2} \cdot 16 = 16 \]

### Step 10: Find K and calculate 4K
Thus, we have:
\[ K = 16 \]
Now, we need to find \( 4K \):
\[ 4K = 4 \times 16 = 64 \]

### Final Answer
The value of \( 4K \) is \( 64 \).

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