Question

[15] 8. (a) Check if the matrix A is defective or not. (b) Use the results of (a) to find the general solution to the system x' = Ax if A = =(24)

Answer 1

A 11 2 Questions Giren, IP 4 a) to check is the matin A k defectue by to use the results of cas to find the general scution to system or not the tema X'=AX. solution: bote? Defective matris A defective malmon & Square matoria cie, nxn matris) that does not have n l'heasly independent eigenvectos a) to check of the materia a la deflective or not we have, A=12 17 now EL 4 x'= ax 1.-1 To find eigenvalues and eigenvestore The characteristic polynomial of the given materie A is giren by, det GA-XI) = 0 12-1 70 4- 09-1304-13+1 =D 8-21-4x+1²+1=0 1²-64+9=0 a-oca-3) = 0 1 = 3,3 crepeated rootsy 90. The eigenvalues are = 3 to 3 The eigenvalue is = 3 with multiplicily 2.) 4 ab Atb

det 27 be the effenwection corresponding to the eigenvalue = 8. ( CA-4I) V=0 12-8 1 11% 4-3] JAD [7 13 -x+y=0 a=y let yol g=1 of y=t, & at, y=t. Vst [J is the eigenvector corresponding to the exgenvallese n- = 8. V=1217 be the eigenvector combespording to the eigenvalue = 3 CA - A IV. = V [2-3 21 4-99 - Ila - -sty=1 y =lta ef 91= let y=t

so, Ve=t 13 418] is the v=[i] the eigenvector corresponding to the eigenvalue 11 A = 3. Here, we could see that The green malonia A bas, eigenvale 3. of multiplicilis 2 but only one distinct, eligenvector creo elgenvectors here ane not lineartes independent). So, the gren mania a iš defective

b) to ford the general duties to the systen X'=AX. using the results of cas, we have, The general solution alt) = Crest [1] + ca est coes [+ []+[:)]