Let A be a square matrix of order 3 such that \( |A|=3 \), then the value of \( \displaystyle\frac{|\mathrm{adj}\,(\mathrm{adj}\,A)|}{|A|} \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 27
For a square matrix of order n, \( |\mathrm{adj}\,A|=|A|^{n-1} \). Applying it twice:
\( \displaystyle |\mathrm{adj}(\mathrm{adj}\,A)|=|\mathrm{adj}\,A|^{n-1} \)
\( \displaystyle =|A|^{(n-1)^2} \)
With n = 3 and |A| = 3:
\( \displaystyle |\mathrm{adj}(\mathrm{adj}\,A)|=3^{4}=81 \)
\( \displaystyle \frac{|\mathrm{adj}(\mathrm{adj}\,A)|}{|A|}=\frac{81}{3}=27 \)
Why the other options are incorrect
Option (2) 81 — Incorrect. 81 is \( |\mathrm{adj}(\mathrm{adj}\,A)| \) itself; it has not been divided by |A| = 3.
Option (3) 3 — Incorrect. 3 is \( \frac{|\mathrm{adj}\,A|}{|A|}=\frac{9}{3} \), using adj only once.
Option (4) 9 — Incorrect. 9 is \( |\mathrm{adj}\,A|=3^2 \), again using adj only once and not dividing.