If \( A \) is a square matrix of order 3 such that \( |A|=3 \), then the value of \( |adj(adjA)| \) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 81
For a square matrix of order n, \( |\mathrm{adj}\,A|=|A|^{n-1} \). Applying this twice:
\( \displaystyle |\mathrm{adj}(\mathrm{adj}\,A)|=|\mathrm{adj}\,A|^{n-1} \)
\( \displaystyle =\left(|A|^{n-1}\right)^{n-1} \)
\( \displaystyle =|A|^{(n-1)^2} \)
With n = 3:
\( \displaystyle |A|^{4}=3^4=81 \)
Why the other options are incorrect
Option (1) 9 — Incorrect. This is |adj A| = 32; the formula has to be applied a second time.
Option (2) 27 — Incorrect. The value is 34 = 81, not 27.
Option (4) 90 — Incorrect. The value is 81, not 90.