If \( A \) is a square matrix of order 3 with \( |A|=4 \), then the value of \( |2\,\mathrm{adj}\,A| \) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 128
For a square matrix B of order n: \( |kB|=k^{n}|B| \) and \( |\mathrm{adj}\,A|=|A|^{n-1} \). Here n = 3.
\( \displaystyle |\mathrm{adj}\,A|=|A|^{2}=16 \)
\( \displaystyle |2\,\mathrm{adj}\,A|=2^{3}\times|\mathrm{adj}\,A| \)
\( \displaystyle |2\,\mathrm{adj}\,A|=8\times 16=128 \)
Tip: Two powers are in play: the scalar comes out as \( k^{n} \), and the adjoint gives \( |A|^{n-1} \). Using \( 2^{2} \) instead of \( 2^{3} \) gives 64, and using \( |A|^{3} \) instead of \( |A|^{2} \) gives 512.