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  3. let ( ain m_6(mathbb{r}) ) be an invertible matrix

Let ( Ain M_6(mathbb{R}) ) be an invertible matrix satisfies ( A^3-4A^2 + 3A =...

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Advance Math

Let ( Ain M_6(mathbb{R}) ) be an invertible matrix satisfies ( A^3-4A^2 + 3A = 0 ) and ( tr(A) = 8. ) Find the characteristics polynomial of A.

 

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Solution

( Ain M_6(mathbb{R}) implies deg(Plambda) )

we have ( hspace{2mm}A^3-4A^2+3A=0 )

( iff Aigg(A^2-4A+3Iigg)=0 )

( iffigg(A-Iigg)igg(A-3Iigg)=0 )

Let ( f(lambda)=igg(lambda-Iigg)igg(lambda-3Iigg) ) Then ( f(A)=0 )

( implies P(lambda)=igg(lambda-1igg)^{n_1}igg(lambda-3igg)^{n_2} hspace{2mm} )where ( n_1+n_2=6 hspace{2mm}(1) )

we have ( tr(A)=8iff n_1(1)+n_2(3)=8 ) ( iff n_1+3n_2=8 )

from.(1) ( egin{cases} n_1+n_2=6 & quad\ n_1+3n_2=8 & quad end{cases} implies 2n_2=2implies n_2=1,n_1=5 )

 

Answer

Therefore. ( P(lambda)=igg(lambda-1igg)^5igg(lambda-3igg) )
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