Let B=[ array ll 1 & 3 1 & 5 array ] and A be a 2 × 2 matrix such that A B^ -1 =A^ -1 . If B C B^ -1 =A and C^4+α C^2+β I=O , then 2 β-α is equal to

Correct answer: D. 10 aligned & BCB^ -1 =A & (BCB^ -1 )(BCB^ -1 )=A · A & BCI CB^ -1 =A^2 & BC^2 B^ -1 =A^2 & B^ -1 (BC^2 B^ -1 ) B=B^ -1 (A.A)B aligned From equation (1) aligned & C^2=A^ -1 · A · B & C^2=B aligned aligned & Also AB^ -1 =A^ -1 & AB^ -1 · A=A^ -1 A=I & A^ -1 (AB^ -1 A)=A^ -1 I & B^ -1 A=A^ -1 aligned Now characteristics equation of C^2 is aligned & |C_2-λ I|=0 & |B-λ I|=0 aligned aligned & | array cc 1-λ & 3 1 & 5-λ array |=0 & (1-λ)(5-1)-3=0 (λ^2-6 λ+5)-3=0 & λ^2-6 λ+2=0 & β^2-6 B+2 I=0 & C^4-6 C^2+2 I=0 & α=-6 & β=2 & 2 β-α=4+6=10 aligned

JEE Main 20249 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 9 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2024 · 9 April · Evening Shift · Mathematics · Matrices

Let B=[ array ll 1 & 3 1 & 5 array ] and A be a 2 × 2 matrix such...

Let

B=\left[

\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}

\right] and

A be a

2 \times 2

matrix such that

A B^{-1}=A^{-1}

. If

B C B^{-1}=A

and

C^4+\alpha C^2+\beta I=O

, then

2 \beta-\alpha

is equal to

Options

Correct answer

D. 10

Step-by-step solution

\begin{aligned} & \mathrm{BCB}^{-1}=\mathrm{A} \\ & \Rightarrow\left(\mathrm{BCB}^{-1}\right)\left(\mathrm{BCB}^{-1}\right)=\mathrm{A} \cdot \mathrm{A} \\ & \Rightarrow \mathrm{BCI} \mathrm{CB}^{-1}=\mathrm{A}^2 \\ & \Rightarrow \mathrm{BC}^2 \mathrm{ B}^{-1}=\mathrm{A}^2 \\ & \Rightarrow \mathrm{B}^{-1}\left(\mathrm{BC}^2 \mathrm{ B}^{-1}\right) \mathrm{B}=\mathrm{B}^{-1} \text{ (A.A)B } \end{aligned}

From equation (1)

\begin{aligned} & \mathrm{C}^2=\mathrm{A}^{-1} \cdot \mathrm{A} \cdot \mathrm{B} \\ & \mathrm{C}^2=\mathrm{B} \end{aligned}

\begin{aligned} & \text{ Also } \mathrm{AB}^{-1}=\mathrm{A}^{-1} \\ & \Rightarrow \mathrm{AB}^{-1} \cdot \mathrm{A}=\mathrm{A}^{-1} \mathrm{ A}=\mathrm{I} \\ & \Rightarrow \mathrm{A}^{-1}\left(\mathrm{AB}^{-1} \mathrm{ A}\right)=\mathrm{A}^{-1} \mathrm{I} \\ & \mathrm{B}^{-1} \mathrm{ A}=\mathrm{A}^{-1} \end{aligned}

Now characteristics equation of

\mathrm{C}^2

is

\begin{aligned} & \left|C_2-\lambda I\right|=0 \\ & |B-\lambda I|=0 \end{aligned}

$\begin{aligned} & \Rightarrow\left|\begin{array}{cc}1-\lambda & 3 \\ 1 & 5-\lambda\end{array}\right|=0 \\ & \Rightarrow(1-\lambda)(5-1)-3=0 \Rightarrow\left(\lambda^2-6 \lambda+5\right)-3=0 \\ & \Rightarrow \lambda^2-6 \lambda+2=0 \\ & \Rightarrow \beta^2-6 \mathrm{ B}+2 \mathrm{I}=0 \\ & \Rightarrow \mathrm{C}^4-6 \mathrm{C}^2+2 \mathrm{I}=0 \\ & \alpha=-6 \\ & \beta=2 \\ & \therefore 2 \beta-\alpha=4+6=10\end{aligned}

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