Question

Let $A$ be a square matrix of order 3 and $|A|=9$. Then $|adj(adj A)|=$ ________.

• A. 6561
• B. 6564
• C. 6569
• D. 8187
• E. 8164

Hint:

$|adj A| = |A|^{n-1}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
This problem requires knowledge of the properties of the determinant and the adjugate (or adjoint) of a square matrix. We need to find the determinant of the adjugate of the adjugate of matrix A.
Step 2: Key Formula or Approach
For any \(n \times n\) square matrix M, we have the following properties:
1. \(|\text{adj}(M)| = |M|^{n-1}\)
2. \(\text{adj}(\text{adj}(A)) = |A|^{n-2} A\)
Using the first formula is more direct for finding the determinant. We will apply it twice.
Step 3: Detailed Explanation
We are asked to find \(|\text{adj}(\text{adj}(A))|\).
Let's use the property \(|\text{adj}(M)| = |M|^{n-1}\).
1. First Application:
Let \(M = \text{adj}(A)\). Substituting this into the formula gives:
\[ |\text{adj}(\text{adj}(A))| = |\text{adj}(A)|^{n-1} \] We are given that A is a square matrix of order 3, so \(n=3\).
\[ |\text{adj}(\text{adj}(A))| = |\text{adj}(A)|^{3-1} = |\text{adj}(A)|^2 \] 2. Second Application:
Now we need to find \(|\text{adj}(A)|\). We use the same property again, this time with \(M=A\).
\[ |\text{adj}(A)| = |A|^{n-1} = |A|^{3-1} = |A|^2 \] 3. Combine the results and substitute the given value.
Substitute the expression for \(|\text{adj}(A)|\) back into our equation from step 1:
\[ |\text{adj}(\text{adj}(A))| = (|A|^2)^2 = |A|^4 \] We are given that \(|A| = 9\).
\[ |\text{adj}(\text{adj}(A))| = (9)^4 \] 4. Calculate the final value.
\[ 9^4 = (9 \times 9) \times (9 \times 9) = 81 \times 81 = 6561 \] Alternatively, \(9^4 = (3^2)^4 = 3^8 = (3^4)^2 = 81^2 = 6561\).
Step 4: Final Answer
The value of \(|\text{adj}(\text{adj}(A))|\) is 6561.