Question

(4) (15 marks) Repeat the Question 2 for the following matrices -3 4 0] 0 0 A -2 30 B 0 -1 0 -8 8 1 0 0 1 ū= 10 = > 3

Answer 1

-3 4 0 A= -2 o 3 8. -8 (I we have I = 1 об 0 to P(x) = det CA-XI) where P() is the characteristic polynanial of il matrix A (A-XI) -3-X 4 3 x SL-8 8 I-X bron det (A- XI) (-3-x) [C3 - x)(1-x)] -4 (-2 (1-x)) (3+x)(3+)(\-) + ((1-2)] was - [19-1) (1-1) + 8 - 8x] - [9 9x - 2 + x3 ] + 8 - 8x - 9+9x + x ² x 3 +8 - 8x - x3 + x²+x-1 Hence P(x) = - - M3++1 -1 which is characteristic polynomial. (II) To find eigen values of A det (A-XI) = 0 → -*-*++2-1) = 0 x²-x²-x+1=0 Then,

& x²(x-1)-1(1-1) 20 (+²1) (x-1)=0 -- | |-رارا = ۸ X eigen values of A. (III) Bacis for the eigenspaces of matrix A. we have = 1, 2 = 1, 3 - 1 Find eigen vectors, FOR = 1 ELS - 404 A-I = A-I 0 -2 2 .8 Now perform rom operations to obtain R3 R3-2R, R2 = R2-12 - 4 4 0 o 0 I do I do 0 o RIFLARI 6 0 0 0 Now, (A-I)x=0 0 0 0 IR x2 0 0 0 o

If we take X₂=t, z = 5 then 2,=t, d2 = t , & = 5 Therefore, + x= Itt lo S. t Lot S 19:30 eigen rectors are Hence, the 1 ero 6 ros bozor me si) For = -1 2 2 AxI= A+I 4 VOSE I los -2 40 Two 8 8 Nou þespose roul Operations, 4 0 7 R₂ = R₂-R, - 2 407 R3 = Ry-4R, 4 0 0 8 2 2 2 0 2 2 4 o ooo Nou Gwap rours 2 and 3 8 2 0 2 0 4 1 - 0 - - O-82 4/4 0 0

(A + I)x=0 SPA-AND Now o 0 -Ź HE O &z=t then Az If we take 2,2 t , Therefore, zt 4 Azt 2 x = th to t - 4 Hence elgen vector is 015 1 for A=1 0-25 B The basis for the eigenspace are 0.5 0.25 (I) Algebraic and geometric multiplicity edgen values 1,1,-1 A.M of x= 1 is a A.M of x=-1 is 1

Non G.M for x=1 is [A-I] .4 4 o -2 2 - 8 Here the rank of this is 2 at so G.M = n- Rank 2 3 - 2 - 1 For x=1 - G.M= 1 For x=-1 ACT [A-XI] -2 4 4 ܢܘ . - 2 8 the rank is 2 Take minor and - 3-2 = 1 so. GM = n-Rank For x= -1 ) G.M=1.

-3 D A = 3 -8 8 1 Matrix A is diagonalizable because its all eigen vectors are different and exist eigon vector are :- 0 and for eigen value 1 for eigenvalue - 1/4 Invertible matrix P will be matrix which Contain eigen vector of matrix A. O P= -- D 14 1. * Condition contain is diagonalise matrix which satisfy P'AP=0 its diagonal element Corrosponding to eigen vector :- 1 80 D 1 - 1