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Determinant nenegativ

 
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rage92



Joined: 14 Feb 2012
Posts: 3

PostPosted: Tue Feb 14, 2012 3:44 pm    Post subject: Determinant nenegativ Reply with quote

Fie A,B\in\mathcal{M}_n(\mathbb{R}) ai AB=0_n.

Dem ca \det\left(A^2+B^2+I_n)\ge 0
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Mateescu Constantin
Newton


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

PostPosted: Tue Feb 14, 2012 4:54 pm    Post subject: Reply with quote

Observam ca \left(A^2+I_n\right)\left(B^2+I_n\right)=A^2B^2+A^2+B^2+I_n=A(AB)B+A^2+B^2+I_n=A^2+B^2+I_n , iar in continuare se foloseste problema binecunoscuta: \det\, \left(X^2+Y^2\right)\, \ge\, 0 , unde X\, ,\, Y\in\mathcal{M}_n\left(\mathbb{R}\right) si XY=YX.
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