Match List-I with List-II
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List-I |
List-II |
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(A) If A is a non-singular matrix of order n, then $\text{|A (adj A)|}$ is equal to |
(I) $|A|^{n-1}$ |
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(B) If A is a non-singular matrix of order n, then $\text{|adj (adj A)|}$ is equal to |
(II) $|A|^{n-2}A$ |
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(C) If A is a non-singular matrix of order n, then $\text{adj (adj A)}$ is equal to |
(III) $|A|^n$ |
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(D) If A is a non-singular matrix of order n, then $\text{|(adj A)|}$ is equal to |
(IV) $|A|^{(n-2)^2}$ |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
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Matching and Explanation
(A) |A (adj A)|
We know: A · adj(A) = |A| In
⟹ |A · adj(A)| = | |A| In | = |A|n
So, (A) → (III)
(C) adj(adj A)
For a non-singular matrix A of order n:
adj(adj A) = |A|n−2 A
So, (C) → (II)
(B) |adj(adj A)|
Taking determinants on both sides of (C):
|adj(adj A)| = | |A|n−2 A |
= |A|(n−2)n · |A|
= |A|(n−2)²
So, (B) → (IV)
(D) |adj A|
We know: |adj A| = |A|n−1
So, (D) → (I)