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Matrice de ordin 2

 
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Mateescu Constantin
Newton


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

PostPosted: Sun Aug 29, 2010 4:11 pm    Post subject: Matrice de ordin 2 Reply with quote

Sa se determine toate matricele A=\left\(\begin{array}{ccc}
a & b \\\\\\\\     
c & d\end{array}\right\)\in\mathcal{M}_2(\mathbb{C}) pentru care A^n=\left\(\begin{array}{cccc}
a^n & b^n \\\\\\\\    
c^n & d^n\end{array}\right\) , \forall\ n\in\mathbb{N}^{\ast} .
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DrAGos Calinescu
Thales


Joined: 07 Dec 2008
Posts: 142
Location: Pitesti

PostPosted: Sun Aug 29, 2010 5:30 pm    Post subject: Reply with quote

Din egalitatea  A=A^2 obtinem  bc=0, b(a+d)=b^2, c(a+d)=c^2
Daca b=0, si c=0 obtinem matricea A=\left\(\begin{array}{ccc} a & 0 \\\\\\\\        0 & b\end{array}\right\) cu a,b\in\mathbb{C} care verifica cerinta(inductie).
Daca b=0 si c\neq 0 obtinem c=a+d.
Trecem la egalitatea A=A^3 unde obtinem ad(a+d)=0 relatie care ne trimite la matricele
A=\left\(\begin{array}{ccc} 0 & 0 \\\\\\\\        a & a\end{array}\right\) A=\left\(\begin{array}{ccc} a & 0 \\\\\\\\        a & 0\end{array}\right\)
Ramane de verificat cazul b\neq 0 si c=0 care se trateaza analog si obtinem matricele
A=\left\(\begin{array}{ccc} 0 & a \\\\\\\\        0 & a\end{array}\right\) A=\left\(\begin{array}{ccc} a & a \\\\\\\\        0 & 0\end{array}\right\)
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