If A is a square matrix of order 3 such that \( |\mathrm{A}|=5 \), then
Match List-I with List-II
| List-I | List-II | ||
| (A) | \( |\mathrm{adj}\,\mathrm{A}| \) | (I) | 125 |
| (B) | \( |3\mathrm{A}| \) | (II) | 135 |
| (C) | \( |\mathrm{A}(\mathrm{adj}\,\mathrm{A})| \) | (III) | 25 |
| (D) | \( |\mathrm{adj}(\mathrm{adj}\,\mathrm{A})| \) | (IV) | 625 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (III), (B) - (II), (C) - (I), (D) - (IV)
For a square matrix A of order n, the standard results on the determinant and the adjoint are
\( \displaystyle |\mathrm{adj\,A}|=|\mathrm{A}|^{n-1} \)
\( \displaystyle |k\mathrm{A}|=k^{n}|\mathrm{A}| \)
\( \displaystyle \mathrm{A}(\mathrm{adj\,A})=|\mathrm{A}|\,\mathrm{I} \)
\( \displaystyle |\mathrm{adj(adj\,A)}|=|\mathrm{A}|^{(n-1)^2} \)
Here n = 3 and |A| = 5.
(A) \( |\mathrm{adj\,A}| \)
This matches (III) 25.
\( \displaystyle |\mathrm{adj\,A}|=5^{3-1} \)
\( \displaystyle =5^2=25 \)
(B) \( |3\mathrm{A}| \)
This matches (II) 135.
\( \displaystyle |3\mathrm{A}|=3^3\times 5 \)
\( \displaystyle =27\times 5=135 \)
(C) \( |\mathrm{A}(\mathrm{adj\,A})| \)
This matches (I) 125.
\( \displaystyle \mathrm{A}(\mathrm{adj\,A})=|\mathrm{A}|\,\mathrm{I}=5\mathrm{I} \)
\( \displaystyle |5\mathrm{I}|=5^3\,|\mathrm{I}|=125 \)
(D) \( |\mathrm{adj(adj\,A)}| \)
This matches (IV) 625.
\( \displaystyle |\mathrm{adj(adj\,A)}|=5^{(3-1)^2} \)
\( \displaystyle =5^4=625 \)
So the matching is (A)-(III), (B)-(II), (C)-(I), (D)-(IV).