Question

3) (9 points) For each of the following matrices Find the eigenvalues and associated eigenvectors. If possible, state the matrices P and D, such that A = PDP-1. (Hint: P is a matrix containing eigenvectors of A on its columns, and D is a diagonal matrix.) If it is not possible to find P and D, just state so. 11-133b a. A = 1 2 2 1-2 -2 -2 2 0 -1 3] b. A = [1 -4 110 0 7 -15 0 1 2 -4. C. A = 3 0 Lo 6 1 0 -21 1 1]

Answer 1

Solver We have A 12 -2 01 2 - 2 To IA-dil zo 1-2 23 find eigen values, let 1 2--2 0 1 1 2 -2-1 -1 = 0 1-2 2 3ad ! => = 1(A-1) (-2) = 0 1:0, 1, 2 To [A-d, I]v=0 find eigen value for 1,=0, let 12 - 20 livil -2 - Ve 1-2 2 3 / luz! = Coo - Vi = Va, Vz =0 v V = 1 Scanned with CamScanner

Similarly for 12 =1, let TA-A, I]v zu 1-2 2 2 / \ vz! → V = 2 V3, V2 = Vz For ts = 2, let TA-d, IJV = 0 To-2 oli vil 2 -4 -1 = V, - į VE, V = 0 , V = -ON Hence 0 0 0 D D = 10 li 2 P = LI Scanned wito CamScanner - 0 0

A = We have lio ol -4 1 2 110-15-4) To find eigen values, let IA-DI! = 0 1-1 -4 Ilo 1Ool 0 7-1 -15 0 2 L so -4-2 | → =) (1-1)2(1-2) = 0 1 =1,1,2. To for 1,=1, let find eigen vector TA-d, I]V = 0 , looo 1 lvl -4 62 110-15-5 ) Vz =) Vis Vin + giving O V7 Scanned with CamScanner

Similarly for 12 = 2, let [A-d, I] V = 0 1-1 0 0 1 lvl 1-4 5 2 | Vi lo ☆ V, 50, V2 = 3 VB V - -21 Hence 112 1 = 372 01 0 -2151 - and 0 Tool loi 0 D = loo 2 Scanned with CamScanner

We have 13 6 -2 | A = 10 11 looi! To find eigen values, we let la-dil=0 3-4 6 -21 lo 01- 1 = = (3-1) (1-2 = 0 1=1,1, 3 To find eigen value for d = 1, let TA-d, I]V = 0 . 1 2 6 -21 iv, olo o il va looo I (vs) оо =) V, - - 3 V2, V = 0 U VE Scanned with CamScanner

Similarly for 1₂ = 3 , let [A-d, I] V = 0 ...lo 6-21 . 1 o 2 vil lo Lo-2 | Vz1 = lo 0 -2 / l vs 0 - 2 lo =) V2=0, Vz = 0 Sime Matrix A has only two linearly independent eigen vators (1) and (3) therefore .. we cannot find p and D Scanned with CamScanner
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