Question

I know A-D. Please do E-G only. Thanks!

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[ 0 ] = W, W_2 is found in part F

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3. (Taken from Boyce & DiPrima) Consider the 3-dimensional system of linear equations Ti 11] X' = AX = 2 1 -1 x 1-3 2 4 (a) Show that the three eigenvalues of the coefficient matrix, A, are 1, = lyd = 2. This is an eigenvalue of multiplicity 3. (b) Show that all the eigenvectors of the coefficient corresponding to the eigenvalue 2) are multiples of r0 = 1 1 - 1 (c) Using the information from parts (a) and (b), find one solution to the system (d) Find a generalized eigenvector, W, of the coefficient matrix, A, by solving (A - 21)W= V Use this generalized eigenvector and the other information you have found to write down a second solution to the system. (e) Show that if V and W are as above, and W, is a vector satisfying (A-21)W =W, then X V + te?'W, +2W, is a solution to the system. (1) Find a vector W, satisfying (A-21)W, = W. and use it to write a third solution to the system. (g) Compute the Wronskian for the three solutions you found in parts (e), (d), and (0) Is it non-zero? Do these three solutions form a fundamental set?

Answer 1

0-3 The given matrix is, x = AX = 11 i 11x . L32 u As given in part (a) x=2 is an eigen value of A with multiplicity 3. Now we find the eigen vectors corresponding to these eigen values, For 1=2 A-QI = 11- 21 11 - 117 2 1-2 -1 - 2 -1 - C L-3 2 4-2 1- 2 al Ra Ra +2R1, R2 => Rg- 3R, No Tino 1 1 Here rank CA-QI) =< 3 C No. of columns) So one variable is free . . -x+y+2=0 9+2=0 - -x-2+2 zo y=-3 x=0 (a is free variable) Let 2=-1 y=1 so the eigen Hector is V - Pol.

Now we find the generalized eigen vedrew, with the help of v. CA-21) W, = v 1-1 1111211 101 10 il 9 Joool | 3. J Ľ"! -x, ty, +3,=0 9, +2,= 1 -30,+1-2,42,20 , = 1-2, x,=1 let 21=1 g, = 0 Go W, = [1 is the another eigen vecton - . Now we find the other eigen vector Wa - with the help of wi. (A-2I) W₂ = WI F ! 1132 10 Il ya 1 Looo![22] [!] -X + 2 + 2a=1 ya tåg 42=-20 - xy +-22 +2e=1 1 xq=-- Let &a=-1, Ya=1 = Fil is the eigen lector.

In this case solution is given as, PGD = ,vete + c(t WntW,) effecerat, 19 Here 1=2 ett X(+ = ci vedt + 52 (tvt wn) ettt az (tºv, ttw, Tuche So according to our need we can choose the values of c,, Q and ca because c,, ce and Ca are arbitrary constants. 1. S. x = text v + text w, test Wo is a soli to the system Now we find the Woonskian of the systern , V = 507 W, -111 , We = F/7 col)-16-1+1)-ICI-D) → of do This system form a fundamental set solution because determinent is non zero these are linearly independent.