Question

If \(A\) is a square matrix of order 3 such that \(|A| = 4\), then the value of \(|\text{adj}(\text{adj } A)|\) is:

• A. \(16\)
• B. \(64\)
• C. \(256\)
• D. \(12\)

Hint:

Memorize the determinant properties of adjoints: \(|\text{adj } A| = |A|^{n-1}\) and \(|\text{adj}(\text{adj } A)| = |A|^{(n-1)^2}\). These provide instant solutions to such problems.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the determinant of a square matrix \(A\) of a specific order, and we need to evaluate the determinant of the adjoint of the adjoint of \(A\).

Step 2: Key Formula or Approach:
For any non-singular square matrix \(A\) of order \(n\), a standard property linking the determinant of its repeated adjoints to the determinant of the original matrix is:
\[ |\text{adj}(\text{adj } A)| = |A|^{(n-1)^2} \]

Step 3: Detailed Explanation:
We are given:
Order of the matrix, \(n = 3\).
Determinant of the matrix, \(|A| = 4\).
Substitute these values into the standard formula:
\[ |\text{adj}(\text{adj } A)| = |A|^{(3-1)^2} \] Evaluate the exponent:
\[ (3 - 1)^2 = 2^2 = 4 \] So the equation simplifies to:
\[ |\text{adj}(\text{adj } A)| = |A|^4 \] Now plug in the value of \(|A|\):
\[ |\text{adj}(\text{adj } A)| = 4^4 \] \[ 4^4 = 4 \times 4 \times 4 \times 4 = 256 \]

Step 4: Final Answer:
The correct choice is (C).