Question

Let $A$ be a $3 \times 3$ matrix such that $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A ))|=12^4$ Then $\mid A ^{-1}$ adj $A \mid$ is equal to

• A. 1
• B. $\sqrt{6}$
• C. 12
• D. $2 \sqrt{3}$

Answer 1

The Correct Option is D

Step 1: Relationship between the Determinants 

We are given the equation:

\[ |A|^{(n-1)^3} = 12^4 \]

Substituting \( n = 3 \) into the equation:

\[ |A|^8 = 12^4 \]

Now, simplify:

\[ |A|^2 = 12 \] \[ |A| = 2\sqrt{3} \]

Step 2: Calculate the Determinant of the Inverse and Adjugate Matrix

We are asked to find the determinant of \( A^{-1} \cdot \text{adj}(A) \), which can be simplified as:

\[ |A^{-1} \cdot \text{adj}(A)| = |A^{-1}| \cdot |\text{adj}(A)| \]

By the properties of matrix determinants, we know that:

\[ |A^{-1}| = \frac{1}{|A|} \] and \[ |\text{adj}(A)| = |A|^{n-1} \]

Substitute these into the equation:

\[ |A^{-1} \cdot \text{adj}(A)| = \frac{1}{|A|} \cdot |A|^{n-1} \]

Given \( n = 3 \), we substitute this value:

\[ |A^{-1} \cdot \text{adj}(A)| = \frac{1}{|A|} \cdot |A|^2 = |A| \]

Since we already know that \( |A| = 2\sqrt{3} \), the result is:

\[ |A^{-1} \cdot \text{adj}(A)| = 2\sqrt{3} \]