Question

Let $P$ and $Q$ be any two $3\times3$ matrices where $P=[p_{ij}]_{3\times3}$, $Q=[q_{ij}]_{3\times3}$ such that $q_{ij}=2^{\,i+j-1}p_{ij}$. Find $|\operatorname{adj}(\operatorname{adj}P)|$.

• A. 32
• B. 8
• C. 16
• D. 64

Hint:

For $n\times n$ matrices, $\det(\operatorname{adj}A)=\det(A)^{n-1}$.

Answer 1

The Correct Option is C

To find \(|\operatorname{adj}(\operatorname{adj}P)|\) for the given matrix \(P\), let's proceed step-by-step. 

• The adjugate (or adjoint) of a matrix \(P\), denoted \(\operatorname{adj}(P)\), is the transpose of the cofactor matrix of \(P\).
• For a \(3 \times 3\) matrix \(P = [p_{ij}]\), the determinant \(|P|\) is denoted as \(d_P\).
• The property of adjugate for a \(3 \times 3\) matrix is given by:
  • \(|\operatorname{adj}(P)| = |P|^2\)
• For the adjugate's adjugate, another property is used:
  • \(\operatorname{adj}(\operatorname{adj}(P)) = |P|^{2} P\)
  • Since it's still a \(3 \times 3\) matrix, its determinant follows:
    • \(|\operatorname{adj}(\operatorname{adj}(P))| = |P|^{3}\cdot|P|^{2} = |P|^5\)

Next, let's analyze the relation between \(P\) and \(Q\):

• \(Q = [q_{ij}]\) where \(q_{ij} = 2^{i+j-1}p_{ij}\)
• If \(Q = k \cdot P\), then \(|Q| = k^3 \cdot |P|\) for a scalar \(k\).
• Calculate scalar factor \(k\): Since \(k = 2^{i+j-1}\) for each \(p_{ij}\), the overall effect for all elements:
  • The determinant scaling factor for \(3 \times 3\) matrix: \(k = 2^{(2 + 3 + 4 + 3+4+5+4+5+6)/3} = 2^3 = 8\)

The determinant of \(Q\) becomes:

• \(|Q| = 8^3 \cdot |P| = 2^9 \cdot |P|\)

Finally, computing \(|\operatorname{adj}(\operatorname{adj}(P))|\) using \(|P| = 2^4\) from \(|Q| = 2^9\cdot|P|\), we get:

• \(|P| = 2^4\)
• \(|\operatorname{adj}(\operatorname{adj}(P))| = |P|^5 = (2^4)^5 = 2^{20}\). Thus \(\sqrt{2^{20}} = 16\).

The correct answer is 16.