Question

(1 point) Suppose A = - (-11, ] Find an invertible matrix P and a diagonal matrix D so that A = PDP-1. Use your answer to find an expression for A6 in terms of P, a power of D, and P-1 in that order. A6 =

Answer 1

A T11 -12 -9 १] The charecteristie foolynomial or Il-2 00 -12 -9-2 20 il det (A - - Z) = 0 (2+9) (-11) +9620 = x-22-99 +96 >>a-222-3=0 => ar 32+2-3=0 -> (2-3)(2+1) 70 Eigen values of A are 3 and 1. Eigen vector corresponding to 3:- v> [m] so that CA-31) 20 89,+8V2=0 (2 27 12V, 712 V220 => V, +4₂20 The eigen rectar be written het va 8 8 - 12 - 12 rector can ) (3)•(0) - "(*)-(3

sponding to se (1) Eigen meclor corres Figen reclor comfoonding to -1:- (A+2) v=0) /12 8 .-22 -8 3*)=63 12v+862 20 => guitar2 = 0 => V2 2 The rector r can be written as va ani 22 3 2 " ( t (. ( 2 -3 2 The eigen vectore corresponding to -lis • Corrsider the diagonal matrin Dz Consider the matrin P= (23) (34 2 CA 1-3 Then py A=PD P! 3 2 -1-1 43)(**) Noco A6= (Popijo PDPY.PDP. Ppp.PDpt.Pop. ppp! PDop-) 06 (129) (4-3)(739 )( * 2 A6 - ap