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Doua matrice de ordin 3 diferite
Posted: Fri Nov 09, 2007 10:59 pm
by Cezar Lupu
Fie \( A, B\in M_{3}(\mathbb{R}) \) doua matrice astfel incat \( A\neq B \). Daca \( A^3= B^3 \) si \( A^{2}B=B^{2}A \), aratati ca matricea \( A^{2}+B^{2} \) nu este inversabila.
Posted: Sat Mar 01, 2008 11:04 pm
by Ciprian Oprisa
\( (A^2+B^2)(A-B)=A^3-A^2B+B^2A-B^3=O \)
Presupunem k \( \det(A^2+B^2)\neq0 \)
\( \Rightarrow rg(A^2+B^2)=n \)
Din inegalitatea lui Sylvester, \( rg(XY)\geq rg{X}+rg{Y}-n \)
\( \Rightarrow 0=rg(A^2+B^2)(A-B)\geq rg(A^2+B^2)+rg(A-B)-n=rg(A-B) \)
\( \Rightarrow rg(A-B)=0 \ \Rightarrow A=B \) -absurd