If A is a square matrix of order 3 such that \( |A|=3 \), then the value of \( |\mathrm{adj}\,A|+|\mathrm{adj}\,(\mathrm{adj}\,A)| \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 90
For a square matrix of order n, the standard results are
\( \displaystyle |\mathrm{adj}\,A|=|A|^{n-1} \)
\( \displaystyle |\mathrm{adj}(\mathrm{adj}\,A)|=|A|^{(n-1)^2} \)
With n = 3 and |A| = 3:
\( \displaystyle |\mathrm{adj}\,A|=3^{2}=9 \)
\( \displaystyle |\mathrm{adj}(\mathrm{adj}\,A)|=3^{4}=81 \)
\( \displaystyle |\mathrm{adj}\,A|+|\mathrm{adj}(\mathrm{adj}\,A)|=9+81=90 \)
Why the other options are incorrect
Option (1) 108 — Incorrect. 108 = 27 + 81 takes \( |\mathrm{adj}\,A|=|A|^{n}=27 \); the power should be n − 1 = 2.
Option (3) 26 — Incorrect. The sum is 9 + 81 = 90, not 26.
Option (4) 74 — Incorrect. The sum is 9 + 81 = 90, not 74.