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  • Let be real numbers. Consider a <img alt="3 \times 3" src="https://entrancecorn

Let \alpha$ and $\beta be real numbers. Consider a 3 \times 3 matrix A such that \mathrm{A^2=3 A+\alpha I}. If \mathrm{A^4=21 A+\beta I}, then

 

Option: 1

\beta=-8
 


Option: 2

\beta=8
 


Option: 3

\alpha=4
 


Option: 4

\alpha=1


Answers (1)

best_answer

\mathrm{A}^2=3 \mathrm{~A}+\alpha \mathrm{I}                  .........(1)

\mathrm{and \: \: A^4=21 A+\beta I}        ...........(2)

\mathrm{Now \: A^4=A^2 \cdot A^2}

\mathrm{A}^4=(3 \mathrm{~A}+\alpha \mathrm{I}) \cdot(3 \mathrm{~A}+\alpha \mathrm{I})     ...........{From(1)}

\mathrm{ A^4=9 A^2+6 \alpha A+\alpha^2 I }              .............(3)

From (2) and (3)
\mathrm{9 A^2+6 \alpha A+\alpha^2 I=21 A+\beta I}
putting value of \mathrm{A}^2 from (1)

\begin{aligned} & 9(3 \mathrm{~A}+\alpha \mathrm{I})+6 \alpha \mathrm{A}+\alpha^2 \mathrm{I}=21 \mathrm{~A}+\beta \mathrm{I} \\ & (27+6 \alpha) \mathrm{A}+\left(9 \alpha+\alpha^2\right) \mathrm{I}=21 \mathrm{~A}+\beta \mathrm{I} \end{aligned}
by comparison

\begin{array}{ll} 27+6 \alpha=21 \text { and } & 9 \alpha+\alpha^2=\beta \\ \Rightarrow 6 \alpha=-6 & \text { putting } \alpha=-1 \\ \Rightarrow \alpha=-1 & \therefore \beta=-8 \end{array}
 

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mansi

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