Given-a-matrix-A-a-b-c-d-satisfies-the-equation-A-2-A-7I-0-where-I-1-0-0-1-Find-the-value-of-trace-of-A-

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

G i v e n a m a t r i x A = (\begin{matrix} a b \\ c d \end{matrix}) s a t i s f i e s

t h e e q u a t i o n A^{2} + λ A + 7 I = 0

w h e r e I = (\begin{matrix} 1 0 \\ 0 1 \end{matrix}) . F i n d t h e v a l u e o f

t r a c e o f A

Answered by bobhans last updated on 19/Sep/20

A^{2} = (\begin{matrix} a b \\ c d \end{matrix}) (\begin{matrix} a b \\ c d \end{matrix}) = (\begin{matrix} a^{2} + b c a b + b d \\ a c + c d b c + d^{2} \end{matrix})

λ A = (\begin{matrix} a λ b λ \\ c λ d λ \end{matrix}) \Rightarrow A^{2} + λ A = (\begin{matrix} - 7 0 \\ 0 - 7 \end{matrix})

\Rightarrow (\begin{matrix} a^{2} + b c + λ a a b + b d + b λ \\ a c + c d + c λ b c + d^{2} + d λ \end{matrix}) = (\begin{matrix} - 7 0 \\ 0 - 7 \end{matrix})

\to c (a + d + λ) = 0; a + d = - λ

\to b (a + d + λ) = 0; a + d = - λ

\Rightarrow a^{2} + b c + λ a = - 7

\Rightarrow a^{2} + b c + a (- a - d) = - 7; b c - a d = - 7

a d - b c = 7; d e t (A) = 7

t r a c e (A) = - λ

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