Question

Let \( A \) be a matrix of order 3 such that \( |A| = -4 \). Then, the value of \( |adj(adj(2adj(A)))^{-1}| \) is:

• A. \( \dfrac{1}{2^{16}} \)
• B. \( \dfrac{1}{2^{28}} \)
• C. \( \dfrac{1}{2^{30}} \)
• D. \( \dfrac{1}{2^{20}} \)

Hint:

For a matrix of order 3, the determinant of \( adj(adj(A)) \) is always \( |A|^4 \). Be extremely careful with the scalar \( k \); it must be raised to the power of \( n \) (the order) before being multiplied by the determinant.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to evaluate the determinant of a complex adjoint expression involving a \(3 \times 3\) matrix \(A\).
Step 2: Key Formula or Approach:
For a matrix of order \(n\):
1. \(|adj(M)| = |M|^{n-1}\)
2. \(|adj(adj(M))| = |M|^{(n-1)^2}\)
3. \(|kM| = k^n |M|\)
4. \(|M^{-1}| = \frac{1}{|M|}\)
Step 3: Detailed Explanation:
Given: \(n = 3\), \(|A| = -4\).
Let \(M = 2adj(A)\).
The expression is \(|adj(adj(M^{-1}))|\).
Using the second formula:
\[ |adj(adj(M^{-1}))| = |M^{-1}|^{(3-1)^2} = |M^{-1}|^4 = \frac{1}{|M|^4} \]
Now, calculate \(|M|\):
\[ |M| = |2adj(A)| = 2^3 |adj(A)| \]
Using formula 1:
\[ |adj(A)| = |A|^{3-1} = |A|^2 = (-4)^2 = 16 \]
So, \(|M| = 8 \times 16 = 128 = 2^7\).
Substitute this into the expression:
\[ \text{Value} = \frac{1}{(2^7)^4} = \frac{1}{2^{28}} \]
Step 4: Final Answer:
The value is \(\frac{1}{2^{28}}\).