If \( A=\begin{bmatrix}1 & 2 & 2\\ 2 & 1 & 2\\ 2 & 2 & 1\end{bmatrix} \), then \( |\mathrm{adj}\,(\mathrm{adj}\,A)| \) is :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 625
For a square matrix of order \( n \), \( |\mathrm{adj}\,A|=|A|^{n-1} \).
Applying this twice gives \( |\mathrm{adj}(\mathrm{adj}\,A)|=|A|^{(n-1)^{2}} \), which is \( |A|^{4} \) for \( n=3 \).
Now expand \( |A| \) along the first row:
\( |A|=1(1-4)-2(2-4)+2(4-2) \)
\( =-3+4+4 \)
\( =5 \)
Therefore
\( |\mathrm{adj}(\mathrm{adj}\,A)|=5^{4}=625 \)
Why the other options are incorrect
Option (1) 25 — Incorrect. This is \( |\mathrm{adj}\,A|=5^{2} \), the formula applied only once.
Option (2) 125 — Incorrect. This is \( 5^{3} \); the power needed is \( (3-1)^{2}=4 \).
Option (4) 5 — Incorrect. This is \( |A| \) itself.