If \( A=\begin{bmatrix}1 & 2 & -1\\-1 & 1 & 2\\2 & -1 & 1\end{bmatrix} \), then \( |adj\ (adj\ A)| \) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 14^4 \)
\( |A|=1(1+2)-2(-1-4)+(-1)(1-2) \)
\( =3+10+1 \)
\( =14 \)
For a matrix of order n, \( |\mathrm{adj}(\mathrm{adj}\,A)|=|A|^{(n-1)^2} \). With n = 3:
\( |\mathrm{adj}(\mathrm{adj}\,A)|=14^{4} \)
Why the other options are incorrect
Option (1) 14 — Incorrect. This is \( |A| \) itself.
Option (2) \( 14^2 \) — Incorrect. This is \( |\mathrm{adj}\,A| \); the formula has to be applied a second time.
Option (3) \( 14^3 \) — Incorrect. The power is \( (3-1)^2=4 \), not 3.