If A is a matrix of order 3 × 3 such that \( |\mathrm{A}|=4 \) and \( |\mathrm{k}\,\mathrm{adj}\,\mathrm{A}|=32|\mathrm{A}|,\ \mathrm{k}\in\mathrm{R} \); then the value of k is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2
For a square matrix of order n: \( |kB|=k^{n}|B| \) and \( |\mathrm{adj}\,A|=|A|^{n-1} \). Here n = 3.
\( \displaystyle |k\,\mathrm{adj}\,A|=k^{3}|\mathrm{adj}\,A| \)
\( \displaystyle =k^{3}|A|^{2} \)
\( \displaystyle =16k^{3} \)
\( \displaystyle 16k^{3}=32\times 4=128 \)
\( \displaystyle k^{3}=8\ \Rightarrow\ k=2 \)
Tip: The scalar k comes out of a determinant of order n as \( k^{n} \), not k; forgetting the power is the usual slip in this type of question.