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: Use standard results of determinants and adjoint

For any non-singular square matrix A of order n:

adj(adj A) = |A|ⁿ⁻² A

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Step 2: Apply the result for a 3 × 3 matrix

Here, P is a 3 × 3 matrix, so n = 3.

adj(adj P) = |P|³⁻² P = |P| · P

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Step 3: Take determinant on both sides

|adj(adj P)| = | |P| · P |

= |P|³ · |P|

= |P|⁴

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Final Answer:

|adj(adj P)| = |P|⁴