Question

In each part of Exercises 5–6, find the characteristic equation, the eigenvalues, and bases for the eigenspaces of the matrix. 5. (a) [ :

Answer 1

1 A = A-dI - 3-d 1-) Ca-d) 3-4d d-80 4A- o satd- d(d-) (a-r) = d S,-1 Elgen valius A = -1, S and tharactisiti egn t) = det (A-dI) eharairacteristiu 8ne) 4x-5 tox bausen spale - N/A-)= R2R-RI 4 - t sol x CA+AI) satis X X4t22o ut 2=- E 1

Th Hte null veutoy ipace f CA+I) unsists basis E Es=N(A-SI) こy d=5 4 4 1-5 LA-SE)= 2 -2 2 - 2 3-F -2 (A-SI) -2 2 2 4-22 he null spacuf (A-SI) tensistsveetor asis

7 2- A-AI 2+d)(2-d )+ 1 2-d - 4-2)+ = a2+3 tharattinis tii e ex)+3)| values A-E =o ) Por d=13 - (A-dI) 2-J31 JE-H (A-J3 I)2 -2-8 - (-2-1B)72=D 2 X = (2-1') 2+ 4(2t13) (4+3) (2tiB) 2ti13 (2tiE) 1 3 M. Bars

A - A-AT 1-1 Charactaristic C()= -x) C(x) 1-2x (-A)2-0 forcigun valu A-AII0 A=1,1 for bases oigun space (A-AI) e ouho X 4 (A-AI)x20a Thus indapindent by X,X ane basis ign space Note: a indufandent variabls ,7,) 2 elemint in bass A -2 1AATI -2 (-A) (-A charattrustic s CCX) C)Itx-2x velus A-AIl-o (1-A=0 Fox sigin d=1,

for 1 -224 AAE= -2 Tthus soleution 1oA-I)x =o satis - 220 Basis Baris