Question

Q5 8 Points Let A = -4 0 -3 2 3 0 -2 -1 3 Q5.1 6 Points Compute PTA (x) (the characteristic polynomial of TA), Write monomials in descending order of their exponent, x^n for 3" and a/b for Example: 5/2 (polynomial of degree 0) -2x+1/3 (polynomial of degree 1) 2x^2-3x+12 (polynomial of degree 2) Enter your answer here give the eigenvalue(s) of TA (in ascending order if multiple exists, separated by a comma and with a blank after the comma. Example: -0.5, 1, 14). Enter your answer here and give a basis for the corresponding eigenspaces.

Answer 1

Page Join is A z 4 2 2 -1 3 3 the characteristic polynom polynomal of TA is given by Arde 3 2 - 1 3 33 -4- -2 2-) -1 3 3. expand with respect to the secondcolumn get -4- (2-2) -L 3 3-1 (2-2) (-4-2)(3-d) te (2-1) (-12-32 +42 +1246) 20 (2-2) (x2 + dat) - - p-d²-81 +12 20 (2nd) (d² + 3d-22 -6)=0 (2nd) (dCat3) - 2(2+3)=0 -(2-) (dr) (-2) -

- 21 d = 2, -3, 2 The eigen vector's are given by for daz A-21) -2 all - 3 -1 ne O w ng Reduce this matrix to Row echelon form r Ret R₂ - I Ri R₂ + I R R. 2 M to ng -d -6n, tout 203 on ang or 30 tag d we have only one equation willy three unkoon take car arbitaly

we have to take 2 variable free, choose Xg 28 dz at than abone is 3x, ts = S 3 Ai to the eigen vector is x= E S. Hok s & ft - 3s TE is tER } the eigen vector for dan (A -(-3) I ) X. 20 (AfIXO 1 O - 2 " 5 - им 3 o 6 ng

Page 9009 R – ₂ – 3 R R₂ R₂ + 3 R -2 m : 5 ПЛА O o ny here 2 so the Rank of thychatrix hree variable we to choose 2030 5x +raz 20 dat ty e Choog e thas &z= -8 -u, - 28 = a - ne = 28 zu n - 28 -28 so the eigen vector in OR 2 2 &

Page so the characteristic polynomial equation is 13-12 -8dt 12 20 So the characteristic Polynomial es d3 - 22 -8d + 12 & the eigen value is -3, 2,2 Baris for the eigenvector is & the 2 1 3 Since n we know that A matrix Anxo i diagona lizable iff dimſeignespace here non is the order of Matrix but here Aix 3x3 matrix 4 there are 3 baris rector in the eigen space so dinleigenspad) 3 dim (eigen space) = order Matrix so A ii of diagonalisable.