Given a non-singular 3x3 matrix \( A \) with \( |A^{-1}| = 24 \). Since \( |A| = \frac{1}{|A^{-1}|} \), we have \( |A| = \frac{1}{24} \).
We aim to compute \( |2A(\operatorname{adj}(3A))| \). Using the property \( \operatorname{adj}(B) = |B|B^{-1} \), we get \( \operatorname{adj}(3A) = |3A|(3A)^{-1} \).
First, calculate \( |3A| \). For a 3x3 matrix, \( |3A| = 3^3|A| = 27|A| \).
Substituting \( |A| = \frac{1}{24} \), we find \( |3A| = 27 \times \frac{1}{24} = \frac{27}{24} \).
Substituting this into the expression for \( \operatorname{adj}(3A) \):
\(\operatorname{adj}(3A) = \frac{27}{24}(3A)^{-1} \).
Now, we evaluate the target determinant:
\(|2A(\operatorname{adj}(3A))| = |2A| \cdot |\operatorname{adj}(3A)| = |2A| \cdot \left|\frac{27}{24}(3A)^{-1}\right| \).
Calculate \( |2A| \):
\(|2A| = 2^3|A| = 8|A| = 8 \times \frac{1}{24} = \frac{8}{24} = \frac{1}{3} \).
Calculate \( \left|\frac{27}{24}(3A)^{-1}\right| \):
\( \left|\frac{27}{24}(3A)^{-1}\right| = \frac{27}{24}\left|(3A)^{-1}\right| = \frac{27}{24} \times \frac{1}{|3A|} = \frac{27}{24} \times \frac{1}{\frac{27}{24}} = \frac{27}{24} \times \frac{24}{27} = 1 \).
Therefore, the final result is:
\(|2A(\operatorname{adj}(3A))| = \frac{1}{3} \times 1 = \frac{1}{3} \).
The value is \(\frac{1}{3}\).