Question

Let m be the set of all 3×3 real matrices which satisfies x^3-3x^2+2x-1 . Is tgen m is compact. Stack exchange

Answer 1

Answer:

Step-by-step explanation:

|A−λI|⟹−λ3+3λ−2⟹(λ−1)2(λ+2)=0=0⋯(i)=0

So the eigen values are given by: λ1=1, λ2=1, λ3=−2.

Now Tr(A)= Sum of eigen values =λ1+λ2+λ3=0, and

det(A)= Product of eigen values =λ1⋅λ2⋅λ3=−2.

Or, you can apply directly Vita's formula in (i) which gives:

Sum of roots of polynomial =Tr(A)=−0−1=0, and

Product of roots of polynomial =det(A)=−−2−1=−2.