Sa se rezolve ecuatia: \( X^n={\begin{pmatrix}2 & 3 \\ 4 & 6\end{pmatrix}} \); \( n\in\mathbb{N}^* \);\( X\in{M_2\left(\mathbb{R}\right) \).
Tuduce Florian, Zalau
Concurs "Teodor Topan" - problema 2
Moderators: Bogdan Posa, Laurian Filip, Beniamin Bogosel, Radu Titiu, Marius Dragoi
\( X^n=\left( \begin{array} 2 & 3 \\ 4 & 6 \end{array}\right) \).
\( \det X^n=(\det X)^n \) si \( \det X^n=0 \Rightarrow\det X=0 \).
Din Cayley-Hamilton si din inductie rezulta ca \( X^n=(\tr(X))^{n-1} X \). Fixand \( X=\left( \begin{array} a & b \\ c & d \end{array}\right) \) obtinem \( \left( \begin{array} 2 & 3 \\ 4 & 6 \end{array}\right)=(a+d)^{n-1} \left( \begin{array} a & b \\ c & d \end{array}\right) \), de unde rezulta valorile variabilelor \( a, b, c, d \).
\( \det X^n=(\det X)^n \) si \( \det X^n=0 \Rightarrow\det X=0 \).
Din Cayley-Hamilton si din inductie rezulta ca \( X^n=(\tr(X))^{n-1} X \). Fixand \( X=\left( \begin{array} a & b \\ c & d \end{array}\right) \) obtinem \( \left( \begin{array} 2 & 3 \\ 4 & 6 \end{array}\right)=(a+d)^{n-1} \left( \begin{array} a & b \\ c & d \end{array}\right) \), de unde rezulta valorile variabilelor \( a, b, c, d \).