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Matrici si determinanti

 
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Visu2412
Euclid


Joined: 22 Mar 2012
Posts: 13

PostPosted: Tue Sep 11, 2012 9:29 am    Post subject: Matrici si determinanti Reply with quote

Buna. Am si eu 2 intrebari.
1. Se dau 2 matrice A si B patratice nesingulare cu det(B)=1 si A+9*A^(-1)*B^2=B si AB=BA.
Aratati ca det(A^3+B^3)=28^n.
am incercat in felul urmator:A^3+B^3=(A+B)(A+EPS*B)(A+EPS^2*B)
A+9*A^(-1)*B^2=A^(-1)(A-3B)(A+3B)=B. De aici nu mai stiu ce sa fac.

2.Fie A si B 2 matrice patratice cu elemente reale. Sa se arate ca tr(A^n(AB-BA))=0, oricare ar fi n, numar natural nenul.

Ma puteti va rog ajuta cu o idee? Multumesc anticipat.
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Claudiu Mindrila
Fermat


Joined: 01 Oct 2007
Posts: 557
Location: Targoviste

PostPosted: Tue Sep 11, 2012 5:25 pm    Post subject: Reply with quote

Pentru 2, se foloseste faptul ca pentru orice matrice avem Tr(A+B)=TrA+TrB si Tr(AB)=Tr(BA)
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elev, clasa a XI-a, C. N. "C-tin Carabella", Targoviste
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Visu2412
Euclid


Joined: 22 Mar 2012
Posts: 13

PostPosted: Tue Sep 11, 2012 7:31 pm    Post subject: Reply with quote

Multumesc Very Happy
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Mateescu Constantin
Newton


Joined: 21 Apr 2009
Posts: 398
Location: Londra/Pitesti

PostPosted: Wed Sep 12, 2012 1:59 pm    Post subject: Reply with quote

Prima problema se regaseste in numarul 6-7-8 din GMB/2012. Din pacate, gazeta continua traditia de a publica probleme gresite, iar problema in cauza nu face exceptie de la aceasta regula ...

Puteti incerca urmatorul contraexemplu (presupunand ca n reprezinta dimensiunea matricei - alt lucru de care nici nu se pomeneste in problema ...) : n=2 cu A=\left(\begin{array}{ccc}
1 & -3 \\ \\ 
3 & 0\end{array}\right) si B=I_2 ...
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Visu2412
Euclid


Joined: 22 Mar 2012
Posts: 13

PostPosted: Thu Sep 13, 2012 9:02 am    Post subject: Reply with quote

Dar pasii pentru abordarea acestui tip de problema sunt buni? Mai voiam sa definesc un polinom f(x)=det(A+xB), iar expresiile sa le scriu in functie de f, insa nu am stiut cum sa il calculez pe f.
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