Let A = [ cases 1 & 0 & 0 0 & α & β 0 & β & α cases ] and | 2A | ^ 3 = 2 ^ 21 where α, β ∈ Z , Then a value of α is

Correct answer: B. 5 Given: A = [ cases 1 & 0 & 0 0 & α & β 0 & β & α cases ] and |2A| ^ 3 = 2 ^ 21 Now, |2A| ^ 3 = 2 ^ 21 ⇒ |2A| = 2 ^ 7 ⇒ 2 ^ 3 |A| = 2 ^ 7 ⇒ |A| = 16 ⇒ α ^ 2 - β ^ 2 = 16 Now, taking value from the option we get, α = 5 as α, β ∈ Z , so for other value of α , β is not a integer.

JEE Main 202429 JanuaryMorning Shift

Exam: JEE Main
Year: 2024
Date: 29 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2024 · 29 January · Morning Shift · Mathematics · Matrices

Let A = [ cases 1 & 0 & 0 0 & α & β 0 & β & α cases ] and | 2A | ^...

Let

A = \left[

\begin{cases}1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha\end{cases}

\right] and

| 2A{|}^{3} = {2}^{21}

where

\alpha, \beta \in Z

, Then a value of

\alpha$ is

Options

A $3$
B $5$
C $17$
D $9$

Correct answer

B. $5$

Step-by-step solution

Given: $A =$ \left[ $\begin{cases}1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha\end{cases}$ \right] $and$ {\left|2A\right|}^{3} = {2}^{21} $Now,$ {\left|2A\right|}^{3} = {2}^{21} $$ ⇒ \left|2A\right| = {2}^{7} $$ ⇒ {2}^{3}\left|A\right| = {2}^{7} $$ ⇒ \left|A\right| = 16 $$ ⇒ {\alpha}^{2} - {\beta}^{2} = 16 $Now, taking value from the option we get,$ \alpha = 5 $as$ \alpha, \beta \in Z $, so for other value of$ \alpha $,$ \beta is not a integer.

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