Let A be a square matrix of order 3 and \( |\mathrm{A}|=-3 \), then \( |\mathrm{adj}\,(3\mathrm{A})| \) is equal to:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( 3^{8} \)
For a square matrix B of order n, two standard results hold: \( |\mathrm{adj}\,B|=|B|^{n-1} \) and \( |kB|=k^{n}|B| \). Here n = 3.
\( \displaystyle |3A|=3^{3}|A| \)
\( \displaystyle =27\times(-3) \)
\( \displaystyle =-3^{4} \)
\( \displaystyle |\mathrm{adj}\,(3A)|=|3A|^{3-1} \)
\( \displaystyle =\left(-3^{4}\right)^{2} \)
\( \displaystyle =3^{8} \)
Tip: For a matrix of order 3, \( |\mathrm{adj}\,B|=|B|^{2} \) can never be negative, so Options (2) and (4) can be rejected at sight.