Question

If A is a 4x4 matrix and $|2A|=64$, $B=adj A$ then $|adj B| =$

• A. $2^{18}$
• B. $2^{36}$
• C. $2^{6}$
• D. $2^{9}$

Hint:

$|adj(adj A)| = |A|^{(n-1)^{2}}$. For $n=4$, the power is $(3)^{2} = 9$.

Answer 1

The Correct Option is B

Step 1: Concept
Property: $|kA| = k^{n}|A|$. For a 4x4 matrix, $|2A| = 2^{4}|A| = 16|A|$.

Step 2: Meaning
Given $16|A| = 64$, we find $|A| = 4 = 2^{2}$.

Step 3: Analysis
$B = adj A \implies |B| = |A|^{n-1} = |A|^{3}$. Then $|adj B| = |B|^{n-1} = (|A|^{3})^{3} = |A|^{9}$.

Step 4: Conclusion
Substitute $|A| = 2^{2}$: $|adj B| = (2^{2})^{9} = 2^{18}$. Final Answer: (A)