If A, B are two non - singular matrices of order 3 and I is an identity matrix of order 3 such that `"AA"^(T)=5I` and `3A^(-)=2A^(T)-Aadj(4B)`, then `|B|^(2)` is equal to (where `A^(T)` and `adj(A)` denote transpose and adjoint matrices of the matrix A respectively )

Options

A. `(7^(3))/(5^(3).4^(6))`
B. `(7^(3))/(5^(3).4^(4))`
C. `(7^(4))/(5^(3).2^(12))`
D. `(5^(6))/(7^(3).2^(10))`

Answer

To solve the problem, we need to analyze the given equations involving matrices A and B. Let's go through the solution step by step.

### Step 1: Understand the Given Conditions
We are given two conditions:
1. \( AA^T = 5I \)
2. \( 3A^{-1} = 2A^T - A \text{adj}(4B) \)

### Step 2: Rearranging the Second Condition
From the second condition, we can rearrange it as follows:
\[ 3A^{-1} + A \text{adj}(4B) = 2A^T \]

### Step 3: Multiply Both Sides by A
Now, we will multiply both sides of the equation by \( A \):
\[ 3I + A^2 \text{adj}(4B) = 2AA^T \]

### Step 4: Substitute \( AA^T \)
From the first condition, we know that \( AA^T = 5I \). Substituting this into the equation gives:
\[ 3I + A^2 \text{adj}(4B) = 2(5I) \]
This simplifies to:
\[ 3I + A^2 \text{adj}(4B) = 10I \]

### Step 5: Isolate the Adjoint Term
Rearranging the equation, we get:
\[ A^2 \text{adj}(4B) = 10I - 3I = 7I \]

### Step 6: Take the Determinant of Both Sides
Taking the determinant of both sides, we have:
\[ \text{det}(A^2 \text{adj}(4B)) = \text{det}(7I) \]
Using properties of determinants, we can split the left side:
\[ \text{det}(A^2) \cdot \text{det}(\text{adj}(4B)) = 7^3 \]

### Step 7: Calculate Determinants
We know that:
\[ \text{det}(A^2) = (\text{det}(A))^2 \]
And for the adjoint:
\[ \text{det}(\text{adj}(4B)) = (\text{det}(4B))^{n-1} = (\text{det}(4B))^{2} \quad \text{(since n=3)} \]
Thus:
\[ \text{det}(4B) = 4^3 \cdot \text{det}(B) = 64 \cdot \text{det}(B) \]
So:
\[ \text{det}(\text{adj}(4B)) = (64 \cdot \text{det}(B))^2 = 4096 \cdot (\text{det}(B))^2 \]

### Step 8: Substitute Back into the Determinant Equation
Now substituting back, we have:
\[ (\text{det}(A))^2 \cdot 4096 \cdot (\text{det}(B))^2 = 343 \]
Where \( 343 = 7^3 \).

### Step 9: Solve for \( |B|^2 \)
We can isolate \( |B|^2 \):
\[ (\text{det}(A))^2 \cdot 4096 \cdot (\text{det}(B))^2 = 343 \]
Now, we need to find \( |B|^2 \):
\[ |B|^2 = \frac{343}{(\text{det}(A))^2 \cdot 4096} \]

### Step 10: Final Calculation
Since \( AA^T = 5I \), we know \( \text{det}(A)^2 = 125 \) (because \( 5^3 = 125 \)).
Thus:
\[ |B|^2 = \frac{343}{125 \cdot 4096} \]

### Conclusion
The final expression for \( |B|^2 \) can be simplified further if needed, but this is the key result.