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

Given a matrix A =  (((a   b)),((c   d)) ) satisfies  the equation A^2 +λA+7I = 0  where I= (((1   0)),((0   1)) ) . Find the value of   trace of A

Answered by bobhans last updated on 19/Sep/20

A^2 = (((a    b)),((c    d)) ) (((a    b)),((c     d)) )= (((a^2 +bc    ab+bd)),((ac+cd   bc+d^2 )) )  λA= (((aλ    bλ)),((cλ    dλ)) ) ⇒A^2 +λA= (((−7    0)),((   0   −7)) )  ⇒ (((a^2 +bc+λa    ab+bd+bλ)),((ac+cd+cλ   bc+d^2 +dλ)) )= (((−7      0)),((   0    −7)) )  → c(a+d+λ) = 0 ; a+d=−λ  →b(a+d+λ)=0 ; a+d =−λ  ⇒a^2 +bc+λa=−7  ⇒a^2 +bc+a(−a−d)=−7 ; bc−ad=−7  ad−bc = 7 ; det(A)=7  trace (A)=−λ