Question

Let A be a non-singular matrix of order 3. If \[ \text{det}\left(3 \text{adj}(2 \text{adj}((\text{det} A) A))\right) = 3^{-13} \cdot 2^{-10} \] and \[ \text{det}\left(3 \text{adj}(2 A)\right) = 2^m \cdot 3^n, \] then \( |3m + 2n| \) is equal to __________.

Answer 1

Correct Answer: 14

We start with the given determinant expression:

\[ |3 \, \text{adj}(2 \, \text{adj}(|A|A))| \]

Now simplifying step-by-step:

\[ |3 \, \text{adj}(2|A|^2 \, \text{adj}A)| \]

\[ |3 \cdot 2^2 \cdot |A|^4 \, \text{adj}(\text{adj}A)| \]

We know that \( |\text{adj}A| = |A|^{n-1} \). Hence, substituting this property, we get:

\[ 3^3 \cdot 2^6 \cdot |A|^{12} \cdot |A|^2 = 3^{-13} \cdot 2^{-10} \]

From this,

\[ |A|^{16} = 3^{-16} \cdot 2^{-16} \]

\[ |A| = \frac{1}{6} \]

Now consider:

\[ |3 \, \text{adj}(2A)| = |3 \cdot 2^2 \, \text{adj}A| = 3^3 \cdot 2^6 |A|^2 \]

\[ = 3 \cdot 2^4 \]

Therefore,

\[ M = 4, \quad n = 1 \]

Hence,

\[ |3m + 2n| = 14 \]

Answer 2

Step 1: Simplify \( |3\operatorname{adj}(2\operatorname{adj}(|A|A))| \)

Using determinant properties:

\[ |3\operatorname{adj}(2\operatorname{adj}(|A|A))| = |3| \times |\operatorname{adj}(2)| \times |\operatorname{adj}(|A|A)|. \]

1. Simplify \( |\operatorname{adj}(|A|A)| \):

Using the property \( |\operatorname{adj}(A)| = |A|^{n-1} \), where \( n = 4 \):

\[ |\operatorname{adj}(|A|A)| = |A|^4. \]

2. Simplify \( |\operatorname{adj}(2)| \):

Using \( \operatorname{adj}(kA) = k^{n-1}\operatorname{adj}(A) \):

\[ |\operatorname{adj}(2)| = 2^{n-1} = 2^3. \]

Thus:

\[ |3\operatorname{adj}(2\operatorname{adj}(|A|A))| = 3^3  \times 2^3  \times |A|^4 \times |A|^4 = 3^3  \times 2^3  \times |A|^8. \]

Given:

\[ |3\operatorname{adj}(2\operatorname{adj}(|A|A))| = 2^{-10}  \times 3^{-13}. \]

Equating powers of \( 2 \) and \( 3 \):

\[ 2^3  \times |A|^8 = 2^{-10}, \quad 3^3  \times |A|^8 = 3^{-13}. \]

Solve for \( |A| \):

\[ |A| = 2^{-1}  \times 3^{-1}. \]

Step 2: Simplify \( |3\operatorname{adj}(2A)| \)

\[ |3\operatorname{adj}(2A)| = |3|  \times |\operatorname{adj}(2A)| = 3^2  \times |\operatorname{adj}(2)|^2  \times |A|^2. \] \[ |3\operatorname{adj}(2A)| = 3^2  \times 2^2 \times (2^{-1}  \times 3^{-1})^2 = 2^4  \times 3^1. \]

Step 3: Compute \( |3m + 2n| \)

Given \( |A| = 2^m \cdot 3^n \), substitute \( m = -4 \), \( n = -1 \):

\[ 3m + 2n = 3(-4) + 2(-1) = -12 - 2 = -14. \] \[ |3m + 2n| = 14. \]