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Let $\mathrm{A=\left(\begin{array}{cc} 1 & 2 \\ -2 & -5 \end{array}\right)}$ . Let $\alpha, \beta \in \mathbb{R}$ be such that $\mathrm{\alpha A^{2}+\beta A=2 I}$ . Then $\mathrm{\alpha+\beta}$ is equal to
Option: 1
$-10$

Option: 2
$-6$

Option: 3
$6$

Option: 4
$10$

Answers (1)

best_answer

Characteristic equation of matric A

$\mathrm{|A-\lambda I|=0} \\$

$\mathrm{\left|\begin{array}{cc} 1-\lambda & 2 \\ -2 & -5-\lambda \end{array}\right|=0} \\$

$\mathrm{\Rightarrow \lambda^{2}+4 \lambda=1} \\$

$\mathrm{\Rightarrow A^{2}+4-A=I} \\$

$\mathrm{\Rightarrow 2 A^{2}+8 A=2 I} \\$ .............(1)

$\mathrm{\text { given that } \alpha A^{2}+\beta A=2 I} \\$ ..............(2)

$\mathrm{\text { comparing equetion } (1)\& \text { (2) we get } }\\$

$\mathrm{\alpha=2, \beta=8} \\$

$\mathrm{\therefore \alpha+\beta=10}$

Hence correct option is 4

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Irshad Anwar

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