Question

Let A =\(\left[\begin{matrix} 2 & 1 & 0 \\   1 & 2 & -1 \\   0 & -1 & 2  \end{matrix} \right]\). If |adj(adj(adj 2A)) | = (16)ⁿ, then n is equal to

• A. 10
• B. 8
• C. 12
• D. 9

Hint:

Use the properties of determinants and the adjugate to simplify higher-order matrix calculations.

Answer 1

The Correct Option is A

We are given the matrix \( A \), and we need to calculate the value of \( n \) in the equation \( \left| \text{adj}(\text{adj}(\text{adj}(2A))) \right| = (16)^n \). 
Step 1: Find the determinant of \( A \).
From the given matrix \( A \), we calculate the determinant \( |A| \): \[ |A| = 2[3] - 1[2] = 4. \] Step 2: Use the properties of the adjugate matrix.
We know the following properties of the adjugate matrix: \[ | \text{adj}(A) | = |A|^{n-1}. \] Therefore, \[ | \text{adj}(\text{adj}(\text{adj}(2A))) | = 2A |(n-1)3 = |2A|^8 = 16^n. \] Step 3: Simplify the equation.
We can now solve the equation: \[ (3^2) 4 = 16n = 16^n. \] Simplifying further: \[ (23 \times 32)^8 = 16^n \quad \Rightarrow 2^{40} = 16^n \quad \Rightarrow 16^{10} = 16^n \quad \Rightarrow n = 10. \] Final Answer: \( n = 10 \).