Question

If $$ A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & -2 \end{bmatrix} $$ then find the value of: $$ |\operatorname{adj}(\operatorname{adj} A)| $$ 
 

• A. \( 16 \)
• B. \( 256 \)
• C. \( 128 \)
• D. \( -256 \)
• E. \( -16 \)

Hint:

For a \( 3 \times 3 \) matrix, remember the shortcut \( |\adj A|=|A|^2 \). If adjoint appears twice, apply the same rule again carefully.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem uses the properties of the adjoint of a matrix and its determinant. We don't need to compute the adjoint matrices themselves.
Step 2: Key Formula or Approach:
For any non-singular square matrix A of order n, we have the following properties:
1. \(|\text{adj}(A)| = |A|^{n-1}\)
2. \(|\text{adj}(\text{adj}(A))| = |A|^{(n-1)^2}\)
Step 3: Detailed Explanation:
The given matrix A is a 3x3 matrix, so the order is \(n=3\).
1. Find the determinant of A, |A|:
The matrix A is an upper triangular matrix (all entries below the main diagonal are zero). The determinant of a triangular matrix is the product of its diagonal elements.
\[ |A| = 1 \times 2 \times (-2) = -4 \] 2. Apply the formula for \(|\text{adj}(\text{adj}(A))|\):
Using the formula with \(n=3\) and \(|A| = -4\):
\[ |\text{adj}(\text{adj}(A))| = |A|^{(3-1)^2} \] \[ = |A|^{2^2} \] \[ = |A|^4 \] Now, substitute the value of the determinant:
\[ |\text{adj}(\text{adj}(A))| = (-4)^4 \] \[ = ((-4)^2)^2 = (16)^2 = 256 \] The determinant is a scalar value, and the result must be non-negative since the exponent is even.
Step 4: Final Answer:
The value of \(|\text{adj}(\text{adj}(A))|\) is 256. Therefore, option (B) is correct.