If A is a square matrix of order 3 × 3 , and \( |\mathrm{adj}\ \mathrm{A}|=25 \), then the value of \( |2\mathrm{A}| \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → ± 40
For a square matrix of order n, \( |\mathrm{adj\,A}|=|\mathrm{A}|^{n-1} \) and \( |k\mathrm{A}|=k^n|\mathrm{A}| \). Here n = 3.
\( \displaystyle |\mathrm{A}|^{2}=25 \)
\( \displaystyle |\mathrm{A}|=\pm 5 \)
\( \displaystyle |2\mathrm{A}|=2^3|\mathrm{A}| \)
\( \displaystyle =8(\pm 5)=\pm 40 \)
Since |adj A| is a square of |A|, the sign of |A| cannot be fixed, so both signs remain.
Why the other options are incorrect
Option (1) 20 — Incorrect. 20 comes from \( |2\mathrm{A}|=2^2|\mathrm{A}| \) with only the positive root; the factor is \( 2^3 \) for a matrix of order 3.
Option (2) ± 20 — Incorrect. ± 20 uses \( 2^2 \) instead of \( 2^3 \).
Option (3) 40 — Incorrect. 40 drops the negative value: \( |\mathrm{A}|^2=25 \) allows \( |\mathrm{A}|=-5 \) as well.