Let P=[p_ i j ] and Q=[q_ i j ] be two square matrices of order 3 such that q_ ij =2^ (i+j-1) p_ ij and det(Q)=2^ 10 . Then the value of det(adj(adj P)) is:

Correct answer: B. 16 vmatrix 2p_ 11 & 2^2p_ 12 & 2^3p_ 13 2^2p_ 21 & 2^3p_ 22 & 2^4p_ 23 2^3p_ 31 & 2^4p_ 32 & 2^5p_ 33 vmatrix = 2^ 10 2^2 · 2 · 2^3 vmatrix p_ 11 & p_ 12 & p_ 13 2p_ 21 & 2p_ 22 & 2p_ 23 2^2p_ 31 & 2^2p_ 32 & 2^2p_ 33 vmatrix = 2^ 10 2^9 vmatrix p_ 11 & p_ 12 & p_ 13 p_ 21 & p_ 22 & p_ 23 p_ 31 & p_ 32 & p_ 33 vmatrix = 2^ 10 |P| = 2 |adj(adj(P))| = |P|^ (n-1)^2 = |P|^4 = 2^4 = 16

JEE Main 202624 JanuaryEvening Shift

Exam: JEE Main
Year: 2026
Date: 24 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2026 · 24 January · Evening Shift · Mathematics · Matrices

Let P=[p_ i j ] and Q=[q_ i j ] be two square matrices of order 3...

Let

P=\left[p_{i j}\right]

and

Q=\left[q_{i j}\right]

be two square matrices of order 3 such that

q_{\mathrm{ij}}=2^{(\mathrm{i}+\mathrm{j}-1)} \mathrm{p}_{\mathrm{ij}}

and

\operatorname{det}(\mathrm{Q})=2^{10}

. Then the value of

\operatorname{det}(\operatorname{adj}(\operatorname{adj} \mathrm{P}))

is:

Options

A81
B16
C124
D32

Correct answer

B. 16

Step-by-step solution

\begin{vmatrix} 2p_{11} & 2^2p_{12} & 2^3p_{13} \\ 2^2p_{21} & 2^3p_{22} & 2^4p_{23} \\ 2^3p_{31} & 2^4p_{32} & 2^5p_{33} \end{vmatrix} = 2^{10}

2^2 \cdot 2 \cdot 2^3 \begin{vmatrix} p_{11} & p_{12} & p_{13} \\ 2p_{21} & 2p_{22} & 2p_{23} \\ 2^2p_{31} & 2^2p_{32} & 2^2p_{33} \end{vmatrix} = 2^{10}

2^9 \begin{vmatrix} p_{11} & p_{12} & p_{13} \\ p_{21} & p_{22} & p_{23} \\ p_{31} & p_{32} & p_{33} \end{vmatrix} = 2^{10} \Rightarrow |P| = 2

|\text{adj}(\text{adj}(P))| = |P|^{(n-1)^2} = |P|^4 = 2^4 = 16

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