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