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: Note the given data.
$A$ is a $3 \times 3$ matrix with $|A| = 4$. We want $|\text{adj}(\text{adj}\,A)|$, the determinant of the adjoint applied twice.
Step 2: Recall the single-adjoint rule.
For an $n \times n$ matrix, $|\text{adj}\,A| = |A|^{\,n-1}$. This is the building block we will use twice.
Step 3: Apply it once.
With $n = 3$, the inner adjoint has determinant \[ |\text{adj}\,A| = |A|^{\,3-1} = |A|^2 \]
Step 4: Apply it again.
Now treat $\text{adj}\,A$ as a new $3 \times 3$ matrix and take its adjoint: \[ |\text{adj}(\text{adj}\,A)| = |\text{adj}\,A|^{\,3-1} = \left(|A|^2\right)^2 = |A|^4 \] This matches the compact formula $|A|^{(n-1)^2} = |A|^{(2)^2} = |A|^4$.
Step 5: Substitute the value.
\[ |\text{adj}(\text{adj}\,A)| = 4^4 \]
Step 6: Compute.
\[ 4^4 = 256 \] which is option (C).
\[ \boxed{|\text{adj}(\text{adj}\,A)| = 256} \]