Question

1. Find the eigenvalues for the following matrices and bases for their corresponding eigenspaces. -28 10 (a) -75 27 -3 -4 6 (b) 8 12-18 4 5 -7 -17 5 5 (c) -40 13 10 -20 5 8

Answer 1

a Given -28 -75 27 subtract & from the diagonal entries of the matrix - 28-X 10 - 75 27 Find determinant of this matrix 1-28-X = 8² + 2-6=0 -75 27-X solve this equation. The roots are 1,=2, da = -3. These are eigenvalues Next find eigenvectors (or eigenspaces. I 1=2 -28-2 -30 -75 27-2 1-75 Reduce this matrix, we get. 10 10 25 Now solve the matrix equation If we take Ve=t, then vi = t V2 =t 1 / 3 23 . Va st 3] t -25 10 10 28-(-3) -75 27-(-3) -75 30 Reduce this matrix - 2/5 - [ 7 -]N- Now, solve the matrix equation - 2/5 CO If we take &q=t, V, = 21, V2=t 45 2t ..V 133.1 +

2,=2 Eigenspace is ½ da =-3, Eigenspace is 25 [

Given G -3 8 4 12 -18 -7 S G subtract & from diagonal entries of matrix. [-3-1 -4 12-) -18 4 5 71 Find determinant of this matrix 1-3-X 8 12-1 -18 = x² +27 ²+1-2=0 4 5 -7X The roots are 2,=2, d2=-1, de 1. There are Eigenvalues. Neat find eigenvectos. I 2 = 2 6 -5 12-2 4 -7-2 -3-2 -4 6 8 -18 8 10 5 5 4 Reduce this matrix 2/3 1 -7/31 0 O 43 1-73 31-10 o .-2/32 7/2 x+2/32 20 x =- -2²2 Y-7/220 = Y = 7/32. z det 2=3 ndo 2 x=4 - 3+1 TI 8 - 4 12 + -18 5 -7+1 6 8 13-28 Le 5-6 Reduce the matrix 4 - O -2 - [-]11-1 x+ 20 = x = -2 Y-22=0 =) Y=22

let 2=1:8= G 4 G 8 4 5 Reduce the matrix V 1 1 -2 0 1 - 2 2 + 1/2=0 = x= -1/22 Y- 22 =D Y = 22 V let z = 2V 22 2 3 Given -17 5 5 -40 13 ID -20 5 8 --17 5 5 --+O 13-2 -20 5 - 3+4x²+3) -18=0 8-x | The roots are &, = -2, I2 =3, 23 =3 These are Eigenvalues. x = -2 : F(-2)-17 5 Cas 5 5 -40 LUD 15 10 13-(-2) 10 8-1- -20 5 -20 5 10 Reduce the matrix o - -2 0 1 Now, solve matern equation ala 0 1 - 2 If we take =t, then viste = 2t, z = t. + V= +

-3-17 5 5 S s 13-3 10 -40 10 - 20 5 8-3 -20 5 5 Reduce this matria - 1/4 - 1/4 Now, solve the matrix equation - 1/4 -14 If we take Ve=t, &z=s, then, + t 2 = t, 3=5. + v= t t S id=-2 =) 2=3 ***