Step 1: Understanding the Concept:
This problem explores the relationship between a matrix, its determinant, and its adjoint. The determinant is a scalar value that provides significant information about a square matrix. One of its most powerful properties is multiplicativity: the determinant of a product of two square matrices of the same order is equal to the product of their individual determinants. Furthermore, the adjoint of a matrix (the transpose of the matrix of cofactors) has a specific determinant property related to the original matrix's determinant and its order. By synthesizing these properties, we can simplify complex matrix expressions into simple numerical calculations.
Step 2: Key Formula or Approach:
The core formulas required for this solution are:
1. The property of the product of determinants: \(|AB| = |A| \cdot |B|\).
2. The property of the determinant of an adjoint matrix: For a matrix \(A\) of order \(n \times n\), \(|\text{adj} A| = |A|^{n-1}\).
3. The fundamental matrix identity: \(A \cdot (\text{adj} A) = |A|I_n\), where \(I_n\) is the identity matrix.
Step 3: Detailed Explanation:
We are given that matrix \(A\) is of order \(n = 3\) and its determinant \(|A| = 3\).
We need to evaluate the determinant of the product \(A(\text{adj} A)\).
Using the multiplication property of determinants:
\[ |A(\text{adj} A)| = |A| \cdot |\text{adj} A| \]
Next, we apply the property of the adjoint determinant for a \(3 \times 3\) matrix (\(n=3\)):
\[ |\text{adj} A| = |A|^{3-1} = |A|^2 \]
Substituting this relationship back into our main expression, we get:
\[ |A(\text{adj} A)| = |A| \cdot |A|^2 = |A|^3 \]
Now, we substitute the known value \(|A| = 3\):
\[ |A(\text{adj} A)| = (3)^3 \]
\[ |A(\text{adj} A)| = 3 \times 3 \times 3 = 27 \]
Alternatively, we can use the identity \(A(\text{adj} A) = |A|I_3\).
Taking the determinant of both sides:
\[ |A(\text{adj} A)| = ||A|I_3| \]
Recall the scalar property of determinants: \(|kM| = k^n|M|\) for a matrix \(M\) of order \(n\). Here, \(k = |A|\) and \(n = 3\):
\[ |A(\text{adj} A)| = |A|^3 \cdot |I_3| \]
Since the determinant of the identity matrix \(|I_3| = 1\):
\[ |A(\text{adj} A)| = |A|^3 \cdot 1 = 3^3 = 27 \]
Step 4: Final Answer:
By applying the properties of determinants and the adjoint of a matrix, we have determined that the value is 27. This matches option (B).