Question

If \( A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & 1 \end{pmatrix} \), then \( |\text{Adj}(A^2)| = \)

• A. 9
• B. 27
• C. 729
• D. 81

Hint:

To solve problems involving determinants of adjugates, remember the key formula: \( |\text{Adj}(M)| = |M|^{n-1} \). Combining this with \(|AB| = |A||B|\) allows you to solve for complex expressions like \(|\text{Adj}(A^k)|\) efficiently without computing the matrices \(A^k\) or their adjugates.

Answer 1

The Correct Option is D

We are asked to find the determinant of the adjugate of \( A^2 \).
We use the property that for any non-singular square matrix \( B \) of order \( n \), we have \( |\text{Adj}(B)| = |B|^{n-1} \).
In this problem, the matrix is \( B = A^2 \) and its order is \( n=3 \).
So, \( |\text{Adj}(A^2)| = |A^2|^{3-1} = |A^2|^2 \).
We also use the property of determinants that \( |A^2| = (|A|)^2 \).
Substituting this into our equation, we get \( |\text{Adj}(A^2)| = ((|A|)^2)^2 = |A|^4 \).
First, we need to calculate the determinant of matrix A.
\( |A| = 1(1 \cdot 1 - 1 \cdot 2) - 2(2 \cdot 1 - 1 \cdot 1) + 2(2 \cdot 2 - 1 \cdot 1) \)
\( |A| = 1(1 - 2) - 2(2 - 1) + 2(4 - 1) \)
\( |A| = 1(-1) - 2(1) + 2(3) \)
\( |A| = -1 - 2 + 6 = 3 \).
Now, we can find the required value:
\( |\text{Adj}(A^2)| = |A|^4 = 3^4 = 81 \).