Question

Find a basis for each eigenspace and calculate the geometric multiplicity of each eigenvalue. 3 2 The matrix A = 0 2 0 has eigenvalues X1 = 2 and X2 1 2 3 For each eigenvalue di, use the rank-nullity theorem to calculate the geometric multiplicity dim(Ex). Find the eigenvalues of A = 0 0 -1 0 0 geometric multiplicity of each eigenvalue. -7- Calculate the algebraic and

Answer 1

3 2 0 2 o The eigen vector of A= given where x = orresponde konding tox, = 2 is is LI 2 3 bH, AX = XY sel X2 23 Mo- 2 2 o JB] - 2 3 2 19, ng x3 34, +242 +23=2n, x+21₂ + z =0(1) 222 =202 x2 2 = x-202 + 343 = 2 3 > 4,-24, +43=0 -(2) (1) + (2) 22, +223 nith = 0 & (1)-(2) => 4x & 3= 2n 3 - 3 = 0 x= 2=0 x2=0 3

O En 3 2 to 252. 190,-1) is a basis of the The eigen vector corresponding to q = 4 is given by AY FX,Y (32 . > 34, +24₂ +dy = 47, 3 = 47, 7-4, + 27, +43 = 0 -(3) 27₂ = 4J₂ 29 ₂ 0 7 Jy = 0 9, +24₂ + 3 343 =143 ²4,24₂ -7 ₂ = 0 y, -4,=0[1:4,39 20 - 2 3 0 20 JE 72 -23 4 & کے عنوان 3 :. Ya (3) {(1,0,1)] is a basis of the eigen space rorresponding to ?2=1. home for the eigene value 2, A – 213 - 3-2 - 2 12-2 o o - -2 3-2 - 2 23 1 2 1 Rq - Ry 2 2 *** О, О