Question

11 18 7 Let 4 6 10 (a) Find the eigenvalues of A. (b) For each eigenvalue find the corresponding eigenvectors. (c) Let 21 and 22 be the eigenvalues of A such that 21 <12. Find a match for 11. Find a match for 12 Find a matching eigenvector vị for 11 - Find a matching eigenvector v2 for 12 Let P and D be 2 x 2 matrices defined as follows: 20 and P = [v1v2] o 22 that is, Vị and v2 are the columns of P Show that PDP 1-4. (d) Simplify (PDP 1). (e) Find D2, D3, D4 and guess D10 (1) Use (c), (d) and (c) to find 410 using only two matrix multiplications Now find a matching answer for each of the following questions. Find a match for 11 Choose.. v Find a match for Az Choose.. Find a match for V1 Choose.. Find a match for V2 Choose.. Simplify (PDP-1) 10 and find a match for the result Find a match for the following matrix or select NO MATCH if you can't find a match. 4093 6138 2046 3068 Choose... Choose... NO MATCH (-2,1) An (-3,2) A20 4090 6140 20403080 PD10 P-1 D10 4 x 2" - 3(1)" 6 x 2" 2( 1)" - 2 x 2" 4( 1)" 6(-1)" 3 x 2" Choose.. 3 x 2"_3(-1)" 5 x 2" 6( 1)" 2( 1)" 2 x 2" 7( 1)" 3 x 2" Choose..

Answer 1

a) Let A = [1 187 na 16 101. To find eigen valikes of A: eigenvalues are roots of characteristic 292_LA-A1 ) 20. let 1A-11 | = |11-9 18 1=0 (11-)(-10-N) – 18 (-6) = 0 > _ -. -110 -2 +13 + 108 CO n --2 0 . > Da II JEDE_4(1)(~ 25 2.00 A = 2, - ingin - m. Eigen' values of A are 2,2-1 42222 b) To find I let v then stre eigenvectors corresponding to a be eigenvector corresponding to 22d. Ava Av G U is nemesi) (A-XI) v=0., Hemi vector , (A - (-1) 1) V=0 (A +I) V do

Date By Rz + (2) Ri li we get 3 120, +18.22=0 14 OV, + ova 20. => qv; +3V2/2011 here are 2 variables & only one eam. - Let us consider Vi are as free variable say Uzat. (&TER) > 200, +3720 = 0,=-34 2 . a y pain. 1-34/27 Lt J If we put t=2. Que get Va ((2) . Now, Eigen vector corresponding to 2222. simillarly, Let to be eigen vector arresponding to 22221 (A - N2I) UZORIAI > 19 [ ] = {:}:

Page No. Date 5) gu,+ 18 U2 -0 * 4, +24220. Let uz as free variable say Uz=r. er be any real number (nonzero) :. 4, +22=0 - UE-28. let ral U=5-27 Hence. eigen vector V corresponding to 2,2-5 & eigenvector u corresponding to 2222 are Fa] 4124 respectively. (c) let pa [Mix] = [V UJ - (2-27 - [ 2 ] - [6]. to show pop-A. 127 L-2 -31. P = 1 127 1P 1 L-2-3). -2-3-279-loni 27 12 ILO 2 JL-2-3 Now, IP - 1 3 1-2 -4 7 12 1 2 JL-2 -3) 11 1-6 18 -10) Hence PDAI-A.

Page No. Date d) simplify (PDA)". Let (PDP)10 = (Ppp) (Ppp.. (PDP') lotimes. L =PD"pt ":plp=I :) 4 * $1= [' :]" = (5) ""} r-10 LO 2 , Gula on Lo oro (:D is diagonal matrix). => D1°= [! 01 lo 1024 -- (PDp") = potopt - POH = [ 4093 L-2046 61387 -3068 (ppp") "* = (4093 51387 4093 L-2046 61387 -3068 J.