27429

27429 (Radu Pop și Vasile Ienutaș, Baia Mare)

Fie $A,B\in M_{3}(\mathbb {R} )$ cu proprietatea că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2 + B^2 = 2AB + BA} . Să se arate că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(4AB - BA) + \det(AB - 4BA) = 10\det(AB + 2BA) - 10\det(2AB + BA). }

Soluție

Avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\det(A + iB)\right|^2 = \det((A - iB)(A + iB)) = \det(A^2 + B^2 + i(AB - BA)) = \det(2AB + BA + i(AB - BA)). }

Fie Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = \det(2AB + BA + X(AB - BA)) \in \mathbb{C}[X]} . Avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = \det(2AB + BA) + \alpha X + \beta X^2 + \det(AB - BA) X^3. }

Cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(i) \in \mathbb{R}} , rezultă $\alpha =\det(AB-BA)$. Din $f(1)=f(-2)=27\det(AB)$, rezultă $\beta =4\det(AB-BA)$. Obținem $f=\det(2AB+BA)+\alpha X+4\alpha X^{2}+\alpha X^{3}.$

Din $\det(AB+2BA)=f(-1)=\det(2AB+BA)+2\alpha$, obținem $\det(AB+2BA)-\det(2AB+BA)=2\alpha$.

Din $\det(4BA-AB)=f(-3)=\det(2AB+BA)+6\alpha ,$