This problem asks us to find the value of the determinant \(|2B(\text{adj}(3A))|\) given the determinants of two \(3 \times 3\) matrices, \(A\) and \(B\). We are given that \(|A| = \frac{1}{2\sqrt{2}}\) and \(|B| = \frac{1}{729}\). The order of the matrices is \(n=3\).

Understanding Determinant Properties

To solve this, we need to use several key properties of determinants:

• The determinant of a product of matrices is the product of their determinants: \(|XY| = |X||Y|\).
• The determinant of a scalar multiple of a matrix: \(|kA| = k^n |A|\), where \(n\) is the order of the matrix.
• The determinant of the adjugate of a matrix: \(|\text{adj}(A)| = |A|^{n-1}\).

Step-by-Step Calculation

Let's break down the calculation of \(|2B(\text{adj}(3A))|\) step by step:

Step 1: Apply the Product Rule

Using the property \(|XY| = |X||Y|\), we can write:

Step 2: Calculate \(|2B|\)

Here, \(k=2\) and the order \(n=3\). Using the scalar multiple property \(|kA| = k^n |A|\):

Substituting the given value \(|B| = \frac{1}{729}\):

Step 3: Calculate \(|\text{adj}(3A)|\)

This requires two sub-steps:

1. Find the determinant of \(3A\): Using the scalar multiple property \(|kA| = k^n |A|\) with \(k=3\) and \(n=3\): \(|3A| = 3^3 |A|\) \(|3A| = 27 |A|\)
2. Find the determinant of the adjugate: Using the property \(|\text{adj}(M)| = |M|^{n-1}\), where \(M = 3A\) and \(n=3\): \(|\text{adj}(3A)| = |3A|^{3-1}\) \(|\text{adj}(3A)| = |3A|^2\)

   Now substitute the expression for \(|3A|\) from the previous step:

   \(|\text{adj}(3A)| = (27 |A|)^2\) \(|\text{adj}(3A)| = 27^2 \times |A|^2\) \(|\text{adj}(3A)| = 729 \times |A|^2\)

   Substitute the given value \(|A| = \frac{1}{2\sqrt{2}}\):

   \(|A|^2 = \left(\frac{1}{2\sqrt{2}}\right)^2 = \frac{1}{(2\sqrt{2})^2} = \frac{1}{4 \times 2} = \frac{1}{8}\)

   So, the determinant of the adjugate is:

   \(|\text{adj}(3A)| = 729 \times \frac{1}{8} = \frac{729}{8}\)

Step 4: Combine the Results

Now substitute the values calculated in Step 2 and Step 3 back into the equation from Step 1:

Step 5: Final Simplification

The terms cancel out:

Therefore, the value of \(|2B(\text{adj}(3A))|\) is 1.