27429

27429 (Radu Pop și Vasile Ienutaș, Baia Mare)

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 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 $\left|\det(A+iB)\right|^{2}=\det((A-iB)(A+iB))=\det(A^{2}+B^{2}+i(AB-BA))=\det(2AB+BA+i(AB-BA)).$

Fie $f=\det(2AB+BA+X(AB-BA))\in \mathbb {C} [X]$. Avem $f=\det(2AB+BA)+\alpha X+\beta X^{2}+\det(AB-BA)X^{3}.$

Cum $f(i)\in \mathbb {R}$, rezultă 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 \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 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(AB + 2BA) = f(-1) = \det(2AB + BA) + 2\alpha} , obținem 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(AB + 2BA) - \det(2AB + BA) = 2\alpha. \tag{1} }

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