Matrice de ordin 2

[Mateescu Constantin]

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\(\begin{array}{cccc}
a^n & b^n \\\\\\\\
c^n & d^n\end{array}\right\) \) , \( \forall\ n\in\mathbb{N}^{\ast} \) .

[DrAGos Calinescu]

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\(\begin{array}{ccc} a & a \\\\\\\\ 0 & 0\end{array}\right\) \)