Step 1: Understanding the Concept:
To evaluate \(|2AA^T|\), we need to use several properties of determinants simultaneously:
- Product Rule: The determinant of the product of two matrices is the product of their determinants (\(|XY| = |X||Y|\)).
- Transpose Rule: The determinant of a matrix is equal to the determinant of its transpose (\(|A^T| = |A|\)).
- Scalar Multiple Rule: For a matrix of order \(n\), \(|kA| = k^n |A|\).
In this problem, the matrix \(A\) is of order 3 (\(n=3\)), and we are given its determinant value.
Step 2: Key Formula or Approach:
The approach involves expanding the expression using properties:
\[ |2AA^T| = 2^n |A \cdot A^T| = 2^3 \times |A| \times |A^T| \]
Step 3: Detailed Explanation:
Let's execute the derivation based on the order \(n = 3\) and \(|A| = -3\):
Step 1: Handle the scalar multiplier '2'.
Since the matrix product \(AA^T\) is of order 3, pulling out the scalar gives:
\[ |2(AA^T)| = 2^3 |AA^T| \]
\[ = 8 |AA^T| \]
Step 2: Apply the product rule for determinants.
\[ |AA^T| = |A| \times |A^T| \]
Step 3: Apply the transpose property.
Since \(|A^T| = |A|\), we can write:
\[ |AA^T| = |A| \times |A| = |A|^2 \]
Step 4: Substitute the numerical value of \(|A|\).
Given \(|A| = -3\):
\[ |AA^T| = (-3)^2 = 9 \]
Step 5: Final calculation.
\[ |2AA^T| = 8 \times 9 = 72 \]
Note: Even though the individual determinant \(|A|\) was negative, the final result for the determinant of the product \(AA^T\) is positive because of the square, and the scalar multiplier \(2^3 = 8\) is also positive.
Step 4: Final Answer:
The value of the determinant is 72.
This matches Option (B).