If \( A=\begin{bmatrix}k & 2\\2 & k\end{bmatrix} \) such that \( |A^3|=125 \) then which of following statements are TRUE?
(A) \( |k\,\mathrm{adj}(3A)|=405 \)
(B) \( |k\,\mathrm{adj}(3A)|=2025 \)
(C) \( |\mathrm{adj}(3A)|=95 \)
(D) \( |\mathrm{adj}(3A)|=45 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (D) only
Since \( |A^3|=|A|^3 \),
\( \displaystyle |A|^3=125\ \Rightarrow\ |A|=5 \)
\( \displaystyle k^2-4=5\ \Rightarrow\ k^2=9 \)
For a square matrix B of order n, \( |\mathrm{adj}\,B|=|B|^{n-1} \) and \( |kB|=k^n|B| \). Here n = 2.
\( \displaystyle |3A|=3^2|A|=9\times 5=45 \)
\( \displaystyle |\mathrm{adj}(3A)|=|3A|^{2-1}=45 \)
\( \displaystyle |k\,\mathrm{adj}(3A)|=k^2\,|\mathrm{adj}(3A)| \)
\( \displaystyle =9\times 45=405 \)
Only \( k^2 \) appears, so k = 3 and k = −3 give the same values.
So \( |\mathrm{adj}(3A)|=45 \) and \( |k\,\mathrm{adj}(3A)|=405 \).
Checking the statements
(A) \( |k\,\mathrm{adj}(3A)|=405 \) — Correct. \( k^2\times 45=9\times 45=405 \).
(B) \( |k\,\mathrm{adj}(3A)|=2025 \) — Incorrect. The value is 405, not 2025.
(C) \( |\mathrm{adj}(3A)|=95 \) — Incorrect. The value is 45, not 95.
(D) \( |\mathrm{adj}(3A)|=45 \) — Correct. For order 2, \( |\mathrm{adj}(3A)|=|3A|=45 \).