Question

Please write the answer to all & uploading problems including this one on the paper and upload the answer in one pdf file after you finish and submit this final 19 points) Let ü - (.:-( Recall that each vector correspond to a point in R' (a). (3 points) Show that the triangle with vertices o, u, jis a equilateral triangle (ie, a triangle with equal-length sides). What is the length of the three sides? (Hint: the third side can be expressed by vector ü - 1.) (b), (3 points) Show that fü+ 1ö is the height of this triangle w.t. the third pide. Le show that ļü + Lüis orthogonal to ü – ö. (c)(3 points) Can you find a pair of vectors ä, ős.t. the triangle with vertex o, a, ő is a equilateral triangle with side-length 1? Please write the answer to all 4 uploading problems including this one on the paper and upload the answer in one pdf file after you finish and submit this final. 0 (13 points) Let A, = 0 1 0 0 (a). (3 points) Find all 3 eigenvalues of A. (The eigenvalues could depend on x.) (b). (7 points) Is A, diagonalizable? If so, diagonalize it. (c). (3 points) For what value of ar is A, not diagonalizable? Explain why.

Answer 1

2 maldix 0 0 - 0 Az لا 0 a) The characteriene equation of que : إدر-م1 | |- 0 1 0 0 1 0 د-۱ 0= (لمحلية -م) ومسلم (م-ل) و دریل در درمحل 0 = 1 - المحلي بمل) 20 (12- 24- 2میل) (لال) ود (كحل + 2A- 2)(م۔ا) المدن العالم تشهد را -۸ تو ان= ة 2( 2421 2 3 eigen values of Ax are d = tx ول=A لال =وهو b) chia he eigenrectory of Ax foul d=1 TI- ہ 0 0 1 0 0 و 0 ہ 0 0 ا ہ ا- 0 1 0 0 0 eare the matrix equation we get

0 oui [.. 0 0 12 0 0 0 V3 we get +B=0 0 O 0 2 V= o o for a = 1+2 1-(1+x) 0 -1 0 21 0-) o 1 - (1+1) 0 -2 0 - 0 0 0 6 |-(1+1) X 0 1 0 0 0 solve the mateix equation we get v Vz=0 v = z 20 V3 0 o V3 0 - 1. 0 7 0 0 0 0 2 0 0 0 0 0 Lox dg = 1-2 1-(1-2) ? 1-(1-1) 1-(1-x) solve une mautix quation, we get Yt2=0 & 42-13 s=0 [w] [-g - O & Vz- Ng

here, number of eigenvectou (say, 3) i equal to the number of wlumns (say, 3) in maltix maltix A li diagonalizable మం. such that A = PDP-1 where P = [v, v₂ v p=0 0 0 0 1 - lai 0 O 1 0 and 0 1+x b = 0 da - o D 23. D D 1-2 0

sor since, mateix Ax has all three distinct eigenvalues except at x=0. A i dicigonalizable for all value of a at 120 12 d = ₂ = ₂ = 1 nautix but we need to check Now, Atxzo then, 0 0 A, 1 0 has 0 D - 0 eigenrecton 0 0 ) V2 - - V = 0 0 sor Au also diagonalizabel x=0 Therefore, there there i no vame of for which a ů not diagonalizable