Question

If \(A\) is a square matrix of order \(n\) such that \(|\operatorname{adj}(\operatorname{adj} A)| = |A|^9\), then the value of \(n\) can be

• A. 4
• B. 2
• C. either 4 or 2
• D. None of these

Hint:

Use \(|\operatorname{adj} A| = |A|^{n-1}\) repeatedly.

Answer 1

The Correct Option is A

To solve the given problem, we need to delve into some properties of determinants and adjoint matrices, specifically for square matrices.

Given:

• \(A\) is a square matrix of order \(n\).
• \(\left|\operatorname{adj}(\operatorname{adj} A)\right| = |A|^9\)

We need to determine the possible value of \(n\).

Concepts Used:

1. The adjoint (or adjugate) of a matrix \(A\), denoted \(\operatorname{adj}(A)\), is defined such that \(A \cdot \operatorname{adj}(A) = |A| \cdot I\), where \(I\) is the identity matrix.
2. If \(A\) is an \(n \times n\) matrix, then the adjoint is \(n \times n\), and it holds that \(\left|\operatorname{adj}(A)\right| = |A|^{n-1}\).
3. \(\operatorname{adj}(\operatorname{adj}(A)) \text{ would similarly have determinant } \left|\operatorname{adj}(A)\right|^{n-1}\).

Step-by-Step Solution:

1. First, recall that \(\left|\operatorname{adj}(A)\right| = |A|^{n-1}\).
2. For the matrix \(\operatorname{adj}(\operatorname{adj}(A))\), its determinant is:
   • \(\left| \operatorname{adj}(\operatorname{adj}(A)) \right| = \left( \left| \operatorname{adj}(A) \right| \right)^{n-1} = \left( |A|^{n-1} \right)^{n-1} = |A|^{(n-1)^2}\)
3. We are given that \(\left|\operatorname{adj}(\operatorname{adj}(A))\right| = |A|^9\).
4. Equating the powers of determinants, we have:
   • \(|A|^{(n-1)^2} = |A|^9\)
5. This implies \((n-1)^2 = 9\).
6. Solving the equation \((n-1)^2 = 9\) gives:
   • \(n-1 = 3\) or \(n-1 = -3\), leading to \(n = 4\) or \(n = -2\).
7. Since \(n\) must be a positive integer, we have \(n = 4\).

Thus, the value of \(n\) is \(4\).

Conclusion:

The correct answer is 4.