JEE Main 2026 Jan 24 Shift 2, Mathematics Q8: Algebra Of Determinants

JEE Main2026Jan 24, Shift 2Mathematics

Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ 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 :

A
32
B
16
C
81
D
124

Solution

Write out $Q$ :

$Q = 2 p_{11} 2^{2} p_{12} 2^{3} p_{13} 2^{2} p_{21} 2^{3} p_{22} 2^{4} p_{23} 2^{3} p_{31} 2^{4} p_{32} 2^{5} p_{33}$

Pull factors out of rows: row 1 has factor $2$ , row 2 has factor $2^{2}$ , row 3 has factor $2^{3}$ , giving $2^{1 + 2 + 3} = 2^{6}$ . From the remaining matrix, pull factors out of columns: column 1 factor $2^{0}$ , column 2 factor $2$ , column 3 factor $2^{2}$ , giving $2^{3}$ .

$det (Q) = 2^{6} \cdot 2^{3} \cdot ∣ P ∣ = 2^{9} ∣ P ∣ = 2^{10} \Rightarrow ∣ P ∣ = 2$ .

For an $n \times n$ matrix, $∣ adj (adj P) ∣ = ∣ P ∣^{(n - 1)^{2}} = ∣ P ∣^{4}$ (here $n = 3$ ).

$∴ ∣ adj (adj P) ∣ = 2^{4} = 16$ .

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JEE Main 2026 Jan 24 Shift 2, Mathematics Q8: Algebra Of Determinants

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