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 $|\operatorname{adj}(\operatorname{adj}P)|$.

Hint:

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

Answer 1

Correct Answer: 16

Step 1: Factor scalar from rows and columns.
\[ |Q|=2^{(1+2+3)+(1+2+3)-3}|P|=2^6|P| \] Step 2: Use adjoint property.
\[ |\operatorname{adj}(\operatorname{adj}P)|=|P|^{(3-1)^2}=|P|^4 \] Step 3: Evaluate.
\[ |P|=2 \Rightarrow |\operatorname{adj}(\operatorname{adj}P)|=2^4=16 \] Final conclusion.
The required value is 16.