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 $A^{2}+B^{2}=2AB+BA$. Să se arate că $\det(4AB-BA)+\det(AB-4BA)=10\det(AB+2BA)-10\det(2AB+BA).$

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 $\alpha =\det(AB-BA)$. 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 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, }