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 step by step, we will follow the given equations and properties of determinants.

### Step 1: Analyze the given equations
We are given two equations:
1. \( AA^T = 3B \)
2. \( 2AB^{-1} = 3A^{-1}B \)

### Step 2: Take determinants of both sides of the first equation
Taking the determinant of both sides of the first equation:
\[ |AA^T| = |3B| \]
Using the property of determinants that states \( |AB| = |A||B| \) and \( |kA| = k^n |A| \) (where \( n \) is the order of the matrix), we have:
\[ |A||A^T| = 3^3 |B| \]
Since \( |A| = |A^T| \), we can write:
\[ |A|^2 = 27|B| \]

### Step 3: Rearranging the equation
From the equation \( |A|^2 = 27|B| \), we can express \( |B| \) in terms of \( |A| \):
\[ |B| = \frac{|A|^2}{27} \]

### Step 4: Take determinants of both sides of the second equation
Now, we take the determinant of both sides of the second equation:
\[ |2AB^{-1}| = |3A^{-1}B| \]
Using the properties of determinants:
\[ 2^3 |A||B^{-1}| = 3^3 |A^{-1}||B| \]
This simplifies to:
\[ 8 |A| \frac{1}{|B|} = 27 \frac{1}{|A|} |B| \]
Rearranging gives:
\[ 8 |A|^2 = 27 |B|^2 \]

### Step 5: Substitute \( |B| \) from Step 3 into this equation
Substituting \( |B| = \frac{|A|^2}{27} \) into \( 8 |A|^2 = 27 |B|^2 \):
\[ 8 |A|^2 = 27 \left(\frac{|A|^2}{27}\right)^2 \]
This simplifies to:
\[ 8 |A|^2 = \frac{27 |A|^4}{729} \]
Multiplying both sides by 729 gives:
\[ 5832 |A|^2 = 27 |A|^4 \]
Dividing both sides by \( |A|^2 \) (assuming \( |A| \neq 0 \)):
\[ 5832 = 27 |A|^2 \]
Thus,
\[ |A|^2 = \frac{5832}{27} = 216 \]

### Step 6: Find \( |B| \)
Now, substituting \( |A|^2 = 216 \) back into \( |B| = \frac{|A|^2}{27} \):
\[ |B| = \frac{216}{27} = 8 \]

### Step 7: Calculate \( \frac{|B|^2}{16} \)
Finally, we need to find \( \frac{|B|^2}{16} \):
\[ |B|^2 = 8^2 = 64 \]
Thus,
\[ \frac{|B|^2}{16} = \frac{64}{16} = 4 \]

### Final Answer
The value of \( \frac{|B|^2}{16} \) is \( 4 \).

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