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Question Number 114554 by bemath last updated on 19/Sep/20

$$GivenamatrixA=\left(\begin{array}{c}ab\\ cd\end{array}\right)satisfies$$$$theequation{A}^2+\lambda A+7I=0$$$$whereI=\left(\begin{array}{c}10\\ 01\end{array}\right).Findthevalueof$$$$traceofA$$

Answered by bobhans last updated on 19/Sep/20

$$A^2=\left(\begin{array}{c}ab\\ cd\end{array}\right)\left(\begin{array}{c}ab\\ cd\end{array}\right)=\left(\begin{array}{c}{a}^2+bcab+bd\\ ac+cdbc+d^2\end{array}\right)$$$$\lambda A=\left(\begin{array}{c}a\lambda b\lambda \\ c\lambda d\lambda \end{array}\right)\Rightarrow A^2+\lambda A=\left(\begin{array}{c}-70\\ 0-7\end{array}\right)$$$$\Rightarrow \left(\begin{array}{c}{a}^2+bc+\lambda aab+bd+b\lambda \\ ac+cd+c\lambda bc+d^2+d\lambda \end{array}\right)=\left(\begin{array}{c}-70\\ 0-7\end{array}\right)$$$$\to c(a+d+\lambda )=0;a+d=-\lambda$$$$\to b(a+d+\lambda )=0;a+d=-\lambda$$$$\Rightarrow a^2+bc+\lambda a=-7$$$$\Rightarrow a^2+bc+a(-a-d)=-7;bc-ad=-7$$$$ad-bc=7;det\left(A\right)=7$$$$trace\left(A\right)=-\lambda$$

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