Question

16 points Save Diagonalize the matrix A = and find an orthogonal matrix P such that P-1AP is diagonal. ot type your solution (work and answer) in the textbox below. Only work on your blank sheets of paper. You will submit your work as one PDF file after submit the test. TTT Paragraph v Arial 3 (12pt) EE. T-

Answer 1

- The characteristic equation of Ais 1 1 - 2 - = (-x)(3-X) - 0 = 0 0 3-x => X = 1,3 let x, be the eigen vector cos responding to 1 AX, = x Then, x - 24 (23) H2 => x = and 284 +372 = > x = 0 and x + x = 0 let 4= e Then, n = 0 & The eigen vector is non-zero real number, let Xz be the eigen rector corresponding to 3. c(-1), where e is an arbitary Then, AX₂ = 342 > (1:)*-(3 => 24 = 32 and 22, +32 = 32 » 29 = 0 and a is any real number. W = d. Then, the eigen rector is d (i); d is an arbitrary real number to. Taking one eigen rector corresponding to each eisen value we get the orthogonal eigen rectors, (-), B= (0) The corresponding oathognormal eigen vectors are va a and ß de

o V2 . Then, P is an orthogonal mabin. Now, te pro mp== ('$) 46a) - salien (14) 3) where 0-663) v2 3 EPD 3 V2 So, D'AP = Deding (1, 3), whose p= fa ci