Sa se rezolve in \( \mathcal{M}_2 \left( \mathbb{C} \right) \) ecuatia \( X^{2}= \left( \begin{array}{cc} 1 & 2 \\ 3 & 6 \end{array} \right) \).
Concursul interjudetean "Teodor Topan", Subiectul I.
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Tudor Micu
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\( \det X=0 \)
Rezulta \( X^2=Tr(X)\cdot X \). Avem \( Tr(X^2)=Tr(Tr(X)\cdot X)=Tr(X)\cdot Tr(X)=Tr(X)^2 \)
\( Tr(X^2)=7 \), rezulta \( Tr(X)=\pm\sqrt{7} \)
Rezulta \( X=\left(\begin{array}{cc}\pm\frac{1}{\sqrt{7}} & \pm\frac{2}{\sqrt{7}} \\ \pm\frac{3}{\sqrt{7}} & \pm\frac{6}{\sqrt{7}} \end{array} \right) \)
Rezulta \( X^2=Tr(X)\cdot X \). Avem \( Tr(X^2)=Tr(Tr(X)\cdot X)=Tr(X)\cdot Tr(X)=Tr(X)^2 \)
\( Tr(X^2)=7 \), rezulta \( Tr(X)=\pm\sqrt{7} \)
Rezulta \( X=\left(\begin{array}{cc}\pm\frac{1}{\sqrt{7}} & \pm\frac{2}{\sqrt{7}} \\ \pm\frac{3}{\sqrt{7}} & \pm\frac{6}{\sqrt{7}} \end{array} \right) \)
Tudor Adrian Micu
Universitatea "Babes Bolyai" Cluj-Napoca
Facultatea de Matematica si Informatica
Universitatea "Babes Bolyai" Cluj-Napoca
Facultatea de Matematica si Informatica