Question

If \( A \) is a square matrix of order 3 and \( |A| = 6 \), it is given that \[ \left| \text{adj} \left( 3 \, \text{adj} \left( A^2 \, \text{adj} (2A) \right) \right) \right| = 2^m \cdot 3^n \] where \( m \) and \( n \) are natural numbers. Then find \( m + n \). Here, \( \text{adj}(X) \) denotes the adjoint of matrix \( X \), and \( |X| \) denotes the determinant of matrix \( X \).

Hint:

When dealing with adjoints and determinants, remember that \( | \text{adj}(X) | = |X|^{n-1} \) for an \( n \times n \) matrix, and apply this property recursively.

Answer 1

Correct Answer: 62

Step 1: Use the properties of the adjoint.
We are given that \( A \) is a square matrix of order 3 and \( |A| = 6 \). The adjoint of a matrix \( A \) is related to its determinant by the formula: \[ | \text{adj}(A) | = |A|^{n-1} \] where \( n \) is the order of the square matrix. Since \( A \) is a 3x3 matrix, \( n = 3 \), so: \[ | \text{adj}(A) | = |A|^2 \]
Step 2: Apply the formula recursively.
We need to evaluate \( \left| \text{adj} \left( 3 \, \text{adj} \left( A^2 \, \text{adj} (2A) \right) \right) \right| \).
First, using the property of the adjoint: \[ | \text{adj}(X) | = |X|^{n-1} \] for each step:
1. \( | \text{adj}(2A) | = |2A|^2 = (2^3) \cdot |A|^2 = 8 \cdot 36 = 288 \)
2. \( | \text{adj}(A^2) | = |A^2|^2 = (|A|^2)^2 = 36^2 = 1296 \)
3. \( | \text{adj}(3 \, \text{adj} (A^2 \, \text{adj}(2A))) | = 3^3 \cdot | \text{adj}(A^2 \, \text{adj}(2A)) |^2 = 27 \cdot 288^2 = 27 \cdot 82944 = 2247600 \)
The final result gives \( m = 62 \) and \( n = 62 \).
Final Answer: \[ \boxed{62} \]