Question

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	If A is a square matrix of order 3 such that \(|A| = 3\), then \(|adj (adj A)| =\)	I.	16
B.	If A is a non-singular matrix and \(|A| = \frac{1}{16}\), then \(|A^{-1}| =\)	II.	512
C.	If A is a square matrix of order 4 such that \(|A| = 2\), then \(|adj (adj A)| =\)	III.	9
D.	If A is a square matrix of order 2 such that \(|A| = 9\), then \(|adj (adj A)| =\)	IV.	81

Choose the correct answer from the options given below:

• A. A - IV, B - I, C - III, D - II
• B. A - IV, B - I, C - II, D - III
• C. A - I, B - IV, C - III, D - II
• D. A - III, B - I, C - IV, D - II

Hint:

Use \(|adj(adj A)| = |A|^{(n-1)^2}\) and \(|A^{-1}| = 1/|A|\).

Answer 1

The Correct Option is B

Step 1: Derive the formula.
From \(A \cdot adj A = |A| I\), taking determinants gives \(|A||adj A| = |A|^n\). So \(|adj A| = |A|^{n-1}\). Now treat \(adj A\) as a new matrix of order n. Then \(|adj(adj A)| = (|A|^{n-1})^{n-1}\).

Step 2: Compute the exponents.
For \(n = 3\), the power is \((3-1)^2 = 4\). For \(n = 4\), it is 9. For \(n = 2\), it is 1.

Step 3: Evaluate each row.
A: \(3^4 = 81\). C: \(2^9 = 512\). D: \(9^1 = 9\). B: the inverse has the reciprocal determinant, so 16.

Step 4: Read off the option.
Values in order A, B, C, D are 81, 16, 512, 9. With the list II labels this is IV, I, II, III.

Final Answer:
Option 2 matches. \[ \boxed{\text{A-IV, B-I, C-II, D-III}} \]