Question

If \( A \) is a square matrix of order 3, then \( | {Adj}({Adj } A^2) | \) is:

• A. \( |A|^2 \)
• B. \( |A|^4 \)
• C. \( |A|^8 \)
• D. \( |A|^{16} \)

Hint:

For any square matrix \( A \) of order \( n \), we have: \[ |{Adj } A| = |A|^{n-1}. \]

Answer 1

The Correct Option is C

Step 1: {Use the determinant property of adjugate matrices}
For a square matrix \( A \) of order \( n \), the determinant of its adjugate is given by: \[ |{adj } A| = |A|^{n-1}. \] Since \( A \) is of order 3, we get: \[ |{adj } A^2| = |A^2|^{3-1} = |A^2|^2. \] Step 2: {Simplify further}
\[ |A^2| = (|A|^2), \] \[ |{adj } A^2| = (|A|^2)^2 = |A|^4. \] Step 3: {Compute \( |{Adj}({Adj } A^2)| \)}
\[ |{Adj}({Adj } A^2)| = (|A|^4)^{3-1} = (|A|^4)^2 = |A|^8. \] Step 4: {Conclusion}
Thus, the correct answer is \( |A|^8 \).

Answer 2

Step 1: Understanding the adjoint of a matrix
For a square matrix \( A \) of order 3, the adjoint of \( A \), denoted \( \text{Adj}(A) \), is the transpose of the cofactor matrix of \( A \). The relationship between the matrix and its adjoint is given by: \[ A \cdot \text{Adj}(A) = \text{Adj}(A) \cdot A = |A| I, \] where \( I \) is the identity matrix and \( |A| \) is the determinant of \( A \).

Step 2: Investigating the adjoint of \( A^2 \)
We are tasked with finding \( | \text{Adj}(\text{Adj}(A^2)) | \). Let's first understand \( \text{Adj}(A^2) \). From the properties of the adjoint and the determinant, we know that: \[ \text{Adj}(A^2) = \left( \text{Adj}(A) \right)^2, \] since \( A^2 \) is a matrix obtained by multiplying \( A \) with itself. This follows from the fact that the adjoint of the product of two matrices is the product of their adjoints: \[ \text{Adj}(A \cdot B) = \text{Adj}(A) \cdot \text{Adj}(B). \]

Step 3: Determinant of the adjoint of a matrix
The determinant of the adjoint of a matrix \( A \) is related to the determinant of \( A \) by the following formula: \[ | \text{Adj}(A) | = |A|^{n-1}, \] where \( n \) is the order of the matrix. For a matrix of order 3, this becomes: \[ | \text{Adj}(A) | = |A|^2. \] Since \( \text{Adj}(A^2) = \left( \text{Adj}(A) \right)^2 \), we can now calculate the determinant of \( \text{Adj}(\text{Adj}(A^2)) \). Using the same property of adjoint determinants: \[ | \text{Adj}(A^2) | = | \left( \text{Adj}(A) \right)^2 | = \left( | \text{Adj}(A) | \right)^2 = \left( |A|^2 \right)^2 = |A|^4. \] Now, we apply the determinant property again: \[ | \text{Adj}(\text{Adj}(A^2)) | = \left( | \text{Adj}(A^2) | \right)^2 = (|A|^4)^2 = |A|^8. \]

Step 4: Conclusion
Thus, the value of \( | \text{Adj}(\text{Adj}(A^2)) | \) is \( |A|^8 \).

The correct answer is: \( |A|^8 \)