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

To solve this problem, we need to determine the value of \( n \) for which \(|\text{adj(adj(adj } 2A))| = 16^n\).

To begin, we first need to understand and calculate some relevant properties of the matrix \( A \) and transformations involving adjugates and determinants.

Step 1: Understand the definition of adjugate

The adjugate (or adjoint) of a matrix is the transpose of the cofactor matrix. If \( A \) is an \( n \times n \) matrix, then:

\[ \text{adj}(A) = \det(A) \times A^{-1} \]

Therefore, \(\text{adj}(cA) = c^{n-1} \det(A) \times A^{-1}\), where \( c \) is a constant.

Step 2: Calculate \( |A| \)

The determinant of \( A = \begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix} \) is calculated as: \[ |A| = 2(2 \times 2 - (-1) \times (-1)) - 1(1 \times 2 - 0 \times (-1)) + 0 = 2(4 - 1) - 2 = 6 - 2 = 4 \]

Step 3: Calculate \(|2A|\)

If \(|A| = 4\), then the determinant of \(2A\) is given by: \[ |2A| = 2^3 \cdot |A| = 8 \times 4 = 32 \]

Step 4: Calculate the determinant of \(\text{adj}(2A)\)

The property of adjugates gives us: \[ \left| \text{adj}(2A) \right| = \left( |2A| \right)^{2} = 32^2 = 1024 \]

Step 5: Calculate \(|\text{adj(adj}(2A))|\)

Applying the adjugate determinant property again: \[ \left| \text{adj(adj}(2A)) \right| = (1024)^2 = 1048576 \]

Step 6: Calculate \(|\text{adj(adj(adj}(2A)))|\)

Using the adjugate property once more gives: \[ \left| \text{adj(adj(adj}(2A))) \right| = (1048576)^2 = 1099511627776 \]

Step 7: Convert to base 16

We now have to express 1099511627776 as a power of 16: \[ 1099511627776 = (16^3)^{10} = 16^{30} \]

Therefore, if \(|\text{adj(adj(adj } 2A)) | = 16^n\), then \( n \) equals 10.

Conclusion: The correct option is 10.