A−λI=⎝⎛−9−λ−8−1643−λ8447−λ⎠⎞
det(A−λI)=∣∣−9−λ−8−1643−λ8447−λ∣∣
=(−9−λ)∣∣3−λ847−λ∣∣−4∣∣−8−1647−λ∣∣
+4∣∣−8−163−λ8∣∣=(−9−λ)(21−10λ+λ2−32)
−4(−56+8λ+64)+4(−64+48−16λ)
=99+11λ+90λ+10λ2−9λ2−λ3
−32−32λ−64−64λ
=−λ3+λ2+5λ+3=0
−λ2(λ+1)+2λ(λ+1)+3(λ+1)=0
−(λ+1)(λ2−2λ−3)=0
−(λ+1)2(λ−3)=0
λ1=−1,λ2=−1,λ3=3
These are the eigenvalues: −1,−1,3.
λ=−1
A−λI=⎝⎛−9+1−8−1643+18447+1⎠⎞
=⎝⎛−8−8−16448448⎠⎞
R2=R2−R1
⎝⎛−80−16408408⎠⎞
R3=R3−2R1
⎝⎛−800400400⎠⎞
R1=R1/(−8)
⎝⎛100−1/200−1/200⎠⎞
⎝⎛100−1/200−1/200⎠⎞⎝⎛x1x2x3⎠⎞=⎝⎛000⎠⎞
If we take x2=t,x3=s, then x1=21t+21s
Thus
x=⎝⎛s/2+t/211⎠⎞s=⎝⎛1/210⎠⎞t+⎝⎛1/201⎠⎞s
The eigenvectors are
⎝⎛1/210⎠⎞,⎝⎛1/201⎠⎞
λ=3
A−λI=⎝⎛−9−3−8−1643−38447−3⎠⎞
=⎝⎛−12−8−16408444⎠⎞
R2=R2−2R1/3
⎝⎛−120−164−8/3844/34⎠⎞
R3=R3−4R1/3
⎝⎛−12004−8/38/344/3−4/3⎠⎞
R3=R3+R2
⎝⎛−12004−8/3044/30⎠⎞
R2=−3R2/8
⎝⎛−12004104−1/20⎠⎞
R1=−R1/12
⎝⎛100−1/310−1/3−1/20⎠⎞
R1=R1+R2/3
⎝⎛100010−1/2−1/20⎠⎞
⎝⎛100010−1/2−1/20⎠⎞⎝⎛x1x2x3⎠⎞=⎝⎛000⎠⎞
If we take x3=t, then x1=21t,x2=21t
Thus
x=⎝⎛t/2t/2t⎠⎞=⎝⎛1/21/21⎠⎞t
The eigenvector is
⎝⎛1/21/21⎠⎞
Eigenvalue: −1, multiplicity: 2, eigenvectors: ⎝⎛1/210⎠⎞,⎝⎛1/201⎠⎞
Eigenvalue: 3, multiplicity: 1, eigenvector: ⎝⎛1/21/21⎠⎞