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

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 $m = -4$, $n = -1$:

$3m + 2n = 3(-4) + 2(-1) = -12 - 2 = -14.$ $|3m + 2n| = 14.$