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Find the eigenvalues and eigenvectors of the following matrices. State the algebraic and geometric multiplicities of each eigenvalue. 1 a. (21) 1 1 2 b. 2 1 1 c. -1 1 4 03 0 0 1

Answer 1

# : n- # @ let A= CIScanned Defn: Let A EMOCR-) → be the nen type matrix and XER". Then (1) Camel A-211=0 -> a be the ergen-value of A (2) AX=2X = (1, x) → be the pair of cigenvalu s cigen vector prounded x 75 (5) Algebraic multipliciday Com): AMCA) = No. of times x is repeated as a root al CACA) (chavadenstic equahun) (47 Gremetne multiplicity Cam): GMCA) dimension of eigen space of a rank (A-11) (2) EMCP) 14-MIN=0 > 12 VEO (2->)(1-2) -2=0 12-31 - 0 X=0 22 = Ŝ (eigen-calves) Ergen-vector for 7,50: CA-2, D3850 (1)-(3) 2x14212=0 take 82=t, then 24 = =yt + X=(*)ti dio Thus 1,20 is eigen calce s Xi = (-) is an eryen-vector, Esgern vector for A2=g; CA-3%)*3 = 0 =) (7-2)) = (3) Rat Ratch (06) (en (3) - Intarz take X22 = t to, then dziet Thus 22=3 is eigen lealce s X2 = (t ;tto ave With CS CamScanner

AM(+1=0)=1 & AM(1₂=3)=1 as they are not repeated GM (210) 2- rank (A-OI) (?!) - 2-1 2- rank 1311-0 CM (22=3) 2- rank (1-31) 2 - Bank (2) as 11h 11= 2220 2-1 - - #(b) let A=(!?) TA-Il co → 2 2 22 241 X12 -2221 +7 (22=0 821 -22220 - 12-29 +(-3) = 0 a=-1, 12-3 Evyen - Vector for 1,2-1: (A +31)x, = = ( ) =) 941+x12=0 x1 = (-1) titto Eigen- vector for 1 = 3 : (A-31 ) X2 → (33)0.) =(:) 2 x=(1)E:HO 0Esgen-ualues 1, , 3 Eigen-veetud Xv = (-1); x = ( 0 G) AM CE 4) = 1, AMX-3) =! A) GM (11=-1) =1; CM (Hz 3) = 1 Since eigenvalues ave dishnet (Hot repeatedl), hence matua Ad diagonalizable to AMA, GMA i) jf is....n CS CamScanner Results: (2) Scanned with

# (6) 1 let A=( • Ergen-calleen 1 4 3 0 IA-411-0 --> Note: 1,= 2,= 1, 73=3 for upper-Aviangular malnx, eigenualues are just diagonal entries For X, Corresponding to 2 =22² | A-11)X = Ō =) 1 1 0 0 + 1 (-03) 33 = 0, -327 13 =0 ; GER take a=t to, then X = ()t; terrifol Pergen-vector for A3= 3 (A-3 I) X3 = 1 -2 4 1,3)-(:) - 20631 -Xze +263 -1/2 X3 = take (33=1 to, -2 2132 +4233 =) 3132 = 24 X31 -1/₂ t to conselts BALIBIRO A-Is o GMC1=1) = 3 Rank (4-1) 2 = = rank(A-1) = 2 alla-El=0 - Resulff: (1) Eigen-velen del (vepeated twice), 13=3 (2) Eigen-vectors X1, X = (!)titto Xz = 13) AMCI:1122, Amldc31c1 Scanned with GM (421)=1 GM(x=3)=1 CS CamScanner kitto -