Question

4 Matrix A is defined as A = [3_21 (a) Find the eigenvalues. (5 marks) (b) Find a corresponding eigenvector for each of the eigenvalues found in (a). (10 marks) (c) Use the above (a) and (b) results to solve the vector-matrix differential equation * = 1} 21x with the initial conditions X(O) = (0) (10 marks)

Answer 1

let 12 Az 32 @ It d be the ergen value of A than; 1A-+1=0 1-1 2 3 2-1 (-) (2-1)-6=0 2-1-21 +126=0 x231-40 A2_4X+X-40 XCA-U+1(1-4)=0 (x-4) CAHITO d24,-1 I values are 114 palergen vector Corresponding to eagen vaker diat rad, Ilv, = 0 soi eigen het vo than 2 3 JIK)10] 3 24, 12H₂ =0 31, +3K2=0 Hg = 4 soi V= Sº, Via

Now lt ₂2 then - 2 1 2 1 1 eigen Vector corresponding de=4 PA-da I / V2 = 0 (213):18) -34, +24,20 392 Le 20 34 = 242 > Y2= 34 y Soi V₂2 کو کو 34 ut Y, 22 Uzz 2 3 so eigen vector Now solving system of diff. egr 1 2 32 Sinai eigen values and vector of matha 1 2 32 are; di = 4; V de 24 i V2> [3]

thom, solenton is goven by Xit) a cent .Vte XIE) = 4,64]+S443) sa et the cht or XIU) 1- -Get +36 64t cusing Initial concletos X 107-10 Kto) = (b) = (443) Now cit2G = 1 -4+39 20 569= Solvay 9-3G = 3:1 3 : 1 G2 Homu solh is X(t) = ² et +2 eht +23 eht or milt) = 3 et 4 oht 4 Me (t) = 3 et te eet