Question

DETAILS LARLINALG8 7.3.033. Show that any two eigenvectors of the symmetric matrix A corresponding to distinct eigenvalues are orthogonal. 3 A = Find the characteristic polynomial of A. |u-A=1 Find the eigenvalues of A. (Enter your answers from smallest to largest.) (14, 12) = Find the general form for every eigenvector corresponding to 1. (Use s as your parameter.) X1 = Find the general form for every eigenvector corresponding to 12. (Use t as your parameter.) X2 = Find x, X2 X1 X2 = Determine whether the eigenvectors corresponding to distinct eigenvalues are orthogonal. x, and x, are orthogonal. x, and x, are not orthogonal.

Answer 1

A-[13] 43 23 Solution Given A=133 33 DI-A= -3 3 3 d-3 III- Al= 3 - (+372-9. 3 2-3 II - Al=0 > (2-3)²-9-o d²+9-6d-9=0 d (d-6)=0 deo and d= 6.00e the eigen valves. (dioda) = (0,6) For the eigen vector [A-d1]X = 0. For, doo, [A-o]x=0. AX=0 B3]K) → 32, +3 X2=0 32, + 342 = 0. Here 22 i's free variable. Subbose 22= Then x = -t. Hence. be the eigen vector. x= [] = [']* For d= 6. [A-61]X=0 37% [0] 3-3

Hence Eigen vector Xi and X2are -> -32,+ 322 30 and 3*, *3*2=0. Let 22= t.. Then. 32,-3tro 32,=t -) 게 Hz) taso eigen vector Noco XuX2= X;X2 = - +2+*2 = 0 X.X2=0 orthogonal.