Question

Let \( |A| = 6 \), where A is a \( 3 \times 3 \) matrix. If \( |\text{adj}(\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^{m \cdot 3^{n} \), then \( m + n \) is equal to :}

Hint:

Remember the rule for scalars in determinants: $|k \cdot A| = k^n |A|$ where $n$ is the order of the matrix. This is the most common place where students lose marks!

Answer 1

Correct Answer: 62

Step 1: Understanding the Concept:
Use the determinant properties of adjoints: \( |\text{adj}(B)| = |B|^{n-1} \). For a \( 3 \times 3 \) matrix, \( n=3 \).
Step 2: Key Formula or Approach:
1. \( |\text{adj}(\text{adj}(B))| = |B|^{(n-1)^2} = |B|^4 \). 2. \( |A^2 \cdot \text{adj}(2A)| = |A|^2 \cdot |2A|^{3-1} = |A|^2 \cdot (2^3|A|)^2 \).
Step 3: Detailed Explanation:
Calculate \( |B| \): \( |B| = |A|^2 \cdot (8|A|)^2 = |A|^2 \cdot 64 |A|^2 = 64 |A|^4 \). Given \( |A| = 6 = 2 \cdot 3 \). \( |B| = 2^6 \cdot (2 \cdot 3)^4 = 2^6 \cdot 2^4 \cdot 3^4 = 2^{10} \cdot 3^4 \). Now find \( |\text{adj}(\text{adj}(B))| = |B|^4 \): \( (2^{10} \cdot 3^4)^4 = 2^{40} \cdot 3^{16} \). Comparing with \( 2^m \cdot 3^n \): \( m = 40, n = 16 \). \( m + n = 40 + 16 = 56 \).
Step 4: Final Answer:
The sum is 56.