Question

Consider the matrix A. [ 300 A = 120 L-6 7 -1 Find the characteristic polynomial for the matrix A. (Write your answer in terms of .) Find the real eigenvalues for the matrix A. (Enter your answers as a comma-separated list.) A= Find a basis for each eigenspace for the matrix A. (smallest eigenvalue) (largest eigenvalue)

Answer 1

The giren matrix is, I soo A=1 i 2o1 Then L67-Il the characteristies pollynomial ab A is det (A-ai) , I is 3x3 identity matrix 13-7 oo = (3-d) (2-2)(--1): [ Expanding writ isf you] = -(XV-52+6) (1+) = -(4-5676 +2²5X+6x). = -(13-474++6) the characteristics pollynomial ab A is [-+ 4 +0.7-6 Therefore The eigen-values at the matrics given by the roots of the equation det (And I)=0 or, (3-1) (2-1) (-4-2)=0 gires, x= 3, 2, 1 A are 8. The real eigen-values of the matrics x= [1, 2, 3 .

tot Ex be the eigen-space correspondings to The ergen value di Ep ={XE1RS: Ax=ax? = x612): (A-11)x=07 , 1,0 are 3*3 identity and zero matrix respectively. hola, En = { XER®: (A+1)x=0} now, (A+1)x=0 W 13007 Troll Hlololl u21=1 11-6 7-1] Lool 0 f4ooTrщТ 1 от 1 isoll u2l=lol 1-670 / no. Lol oy T 424 7 TOT S on t with = -644+12) which gives 41 5o => 1=0 24+ 362 =0 => 12 = 0 com W=0) and us=t arbitrary span which is the basis for E - spona 5:1} „which is the basis for E-1

ар For The ergenvalue &=2 E) = {x ttp® : (А-2:)x=o _нэ (R-2J)x=0. - Г-а о отсут 2-4 о 11 м2 = 1 - 6 1 -2-|] [ 42 ] 1 ТЪТ А тоо || 2 = -61-31 и То Гот |-64 ++-38 ) от » и which avg, 14 = 0, - 64 + 4 4-иQ = 0 25 +2 -3 =0 => 4 . = 2 Kaut ng = + Carbitrarily) gives K2= t. \ \ , : {l is the basis af Eg

For The eigenvalue a=3. Egy = {x&p9: (A-31)x=0} Now, (A-31)X = 0 [ 3-3 oo Tat tot I 2-3 o 11 U2=lolo L-6 7 -1-3/U3] Lol JooolTMT rol m-k2 . 1-614 77 424ng lol which gives molle =0 and 64 + F2-443=0 If we take Wo=t Carbitrany) Then we have 14=42=47: rut rut 142461417 113]. Lt JJH . {[1} is the basis of Eg

Therefore, Basis of the eigm space E, (corresponding to a=-1) eigen-space E2 For The ( corresponds to 2=2) for the eigen-space Eg ( corresponding to 1=3)