Prueba de verdadero/falso: $A\in\mathbb C^{2\times 2}$ es nilpotente. Entonces $A^2=0.$

Question

Prueba de verdadero/falso: $A\in\mathbb C^{2\times 2}$ es nilpotente. Entonces $A^2=0.$

Mi intento:

Cierto: $A^k=0$ para algunos $k\in \mathbb Z^+\\\implies m_A(x)|x^k\\\implies x\text{ is the only irreducible factor of }m_A\\\implies x\text{ is the only irreducible factor of }\chi_A(\text{since }\chi_A\text{ and }m_A\text{ have same irreducible factors})\\\implies\chi_A=x^2\\\implies A^2=0$

¿Estoy en lo cierto?

Answer 1

Tu respuesta es:

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