If \( A \) is a square matrix of order \( 3\times 3 \) and \( |adj\ A|=4 \), then
Match List-I with List-II
| List-I | List-II |
| (A) \( |A^{-1}| \) | (I) 108 |
| (B) \( |3\,adj\,A| \) | (II) \( \pm\displaystyle\frac{1}{2} \) |
| (C) \( |2A| \) | (III) \( \pm 8 \) |
| (D) \( |A.adjA| \) | (IV) \( \pm 16 \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (II), (B) - (I), (C) - (IV), (D) - (III)
For a matrix of order n = 3, \( |\mathrm{adj}\,A|=|A|^{2} \). So \( |A|^2=4 \) and \( |A|=\pm 2 \).
(A) \( |A^{-1}| \)
This matches (II) ±1/2.
\( \displaystyle |A^{-1}|=\frac{1}{|A|}=\pm\frac{1}{2} \)
(B) \( |3\,adj\,A| \)
This matches (I) 108. Using \( |kB|=k^3|B| \) for a 3 × 3 matrix B:
\( \displaystyle |3\,\mathrm{adj}\,A|=27\times 4=108 \)
(C) \( |2A| \)
This matches (IV) ±16.
\( \displaystyle |2A|=2^3|A| \)
\( \displaystyle =8\times(\pm 2)=\pm 16 \)
(D) \( |A.adjA| \)
This matches (III) ±8. Since \( A\,(\mathrm{adj}\,A)=|A|\,I \):
\( \displaystyle |A\,\mathrm{adj}\,A|=|A|^3 \)
\( \displaystyle =(\pm 2)^3=\pm 8 \)
So the matching is (A) - (II), (B) - (I), (C) - (IV), (D) - (III).