Question

If A is a square matrix of order 3 such that |A|= 2, then the value of |adj(adj A)| is:

• A. 2
• B. 4
• C. 8
• D. 16

Answer 1

The Correct Option is D

To find the value of |adj(adj A)|, we utilize some properties of determinants and adjugates.

Given that A is a square matrix of order 3 and |A| = 2, we need to recall that for any n x n matrix A:

• The adjugate (adj) of A, denoted as adj A, satisfies:
  A * adj A = |A| * I, where I is the identity matrix of the same order.
• The determinant of adj A is given by the formula:
  |adj A| = |A|^(n-1), where n is the order of the square matrix.
• For |adj(adj A)|, the formula is:
  |adj(adj A)| = |A|^((n-1)^2).

Since A is a 3x3 matrix, n = 3. Thus, we have:

• |adj A| = |A|^(3-1) = |A|^2 = 2^2 = 4
• |adj(adj A)| = |A|^((3-1)^2) = |A|^4 = 2^4 = 16

Therefore, the value of |adj(adj A)| is 16.