For some α, β ∈ R , let A=[ array ll α & 2 1 & 2 array ] and B=[ array ll 1 & 1 1 & β array ] be such that A^ 2 -4 A+2 I=B^ 2 -3 B+I=O . Then (det (adj(A^ 3 -B^ 3 )))^ 2 is equal to \_\_\_\_ .

Correct answer: 0 From A^2 - 4A + 2I = O : characteristic equation is λ^2 - 4λ + 2 = 0 . For A : α + 2 = 4 α = 2 . From B^2 - 3B + I = O : characteristic equation is λ^2 - 3λ + 1 = 0 . For B : β + 1 = 3 β = 2 . A^3 = 14A - 8I , B^3 = 8B - 3I . A^3 - B^3 = 14A - 8B - 5I = bmatrix 15 & 20 6 & 7 bmatrix . (A^3 - B^3) = 105 - 120 = -15 . For 2×2 matrix: (adj(M)) = (M) . ( (adj(A^3 - B^3)))^2 = 225 .

JEE Main 202621 JanuaryMorning Shift

Exam: JEE Main
Year: 2026
Date: 21 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2026 · 21 January · Morning Shift · Mathematics · Matrices

For some α, β ∈ R , let A=[ array ll α & 2 1 & 2 array ] and B=[...

For some

\alpha, \beta \in \mathbf{R}

, let

A=\left[

\begin{array}{ll}\alpha & 2 \\ 1 & 2\end{array}

\right] and

B=\left[

\begin{array}{ll}1 & 1 \\ 1 & \beta\end{array}

\right]

be such that

A^{2}-4 A+2 I=B^{2}-3 B+I=O

. Then \left(\operatorname{det}

\left(\operatorname{adj}\left(A^{3}-B^{3}\right)\right)\right)^{2} is equal to

\_\_\_\_

Correct answer

Step-by-step solution

From

A^2 - 4A + 2I = O

: characteristic equation is

\lambda^2 - 4\lambda + 2 = 0

.
For

A

\alpha + 2 = 4 \Rightarrow \alpha = 2

.
From

B^2 - 3B + I = O

: characteristic equation is

\lambda^2 - 3\lambda + 1 = 0

.
For

B

\beta + 1 = 3 \Rightarrow \beta = 2

A^3 = 14A - 8I

B^3 = 8B - 3I

A^3 - B^3 = 14A - 8B - 5I = \begin{bmatrix} 15 & 20 \\ 6 & 7 \end{bmatrix}

\det(A^3 - B^3) = 105 - 120 = -15

.
For 2×2 matrix:

\det(\text{adj}(M)) = \det(M)

(\det(\text{adj}(A^3 - B^3)))^2 = 225

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