If A and B are square matrices of order 3 such that `"AA"^(T)=3B` and `2AB^(-1)=3A^(-1)B`, then the value of `(|B|^(2))/(16)` is equal to

Answer

To solve the problem, we need to analyze the given equations involving the square matrices \( A \) and \( B \).

### Step 1: Start with the first equation
Given:
\[ AA^T = 3B \]
Taking the determinant on both sides:
\[ |AA^T| = |3B| \]

### Step 2: Apply properties of determinants
Using the property of determinants, we have:
\[ |AA^T| = |A||A^T| = |A|^2 \]
And for the right side:
\[ |3B| = 3^3 |B| = 27|B| \]
Thus, we can equate:
\[ |A|^2 = 27|B| \tag{1} \]

### Step 3: Move to the second equation
Now consider the second equation:
\[ 2AB^{-1} = 3A^{-1}B \]
Taking the determinant on both sides:
\[ |2AB^{-1}| = |3A^{-1}B| \]

### Step 4: Apply properties of determinants again
Using the properties of determinants:
\[ |2AB^{-1}| = 2^3 |A||B^{-1}| = 8|A|\frac{1}{|B|} = \frac{8|A|}{|B|} \]
And for the right side:
\[ |3A^{-1}B| = 3^3 |A^{-1}||B| = 27\frac{1}{|A|}|B| = \frac{27|B|}{|A|} \]
Thus, we can equate:
\[ \frac{8|A|}{|B|} = \frac{27|B|}{|A|} \tag{2} \]

### Step 5: Cross-multiply to solve for determinants
Cross-multiplying equation (2):
\[ 8|A|^2 = 27|B|^2 \]
Now, we can express \( |A|^2 \) in terms of \( |B|^2 \):
\[ |A|^2 = \frac{27}{8}|B|^2 \tag{3} \]

### Step 6: Substitute equation (3) into equation (1)
Substituting equation (3) into equation (1):
\[ \frac{27}{8}|B|^2 = 27|B| \]
Dividing both sides by 27:
\[ \frac{1}{8}|B|^2 = |B| \]
Multiplying both sides by 8:
\[ |B|^2 = 8|B| \]
Rearranging gives:
\[ |B|^2 - 8|B| = 0 \]
Factoring out \( |B| \):
\[ |B|(|B| - 8) = 0 \]
Thus, \( |B| = 0 \) or \( |B| = 8 \).

### Step 7: Find the required value
We need to find:
\[ \frac{|B|^2}{16} \]
If \( |B| = 8 \):
\[ \frac{|B|^2}{16} = \frac{8^2}{16} = \frac{64}{16} = 4 \]

### Conclusion
The value of \( \frac{|B|^2}{16} \) is:
\[ \boxed{4} \]