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For positive definite matrices $A$ and $C$, positive semidefinite matrices $B$ and $D$. Note that $A=(B+D+eI)^{-1}$, $C=(B+D+fI)^{-1}$ where $I$ is the identity matrix, $e$ and $f$ are postive numbers. It is obvious that under this setting $tr\{BD\}=0$ implies $tr\{ABCD\}=0$. However I want to know whether $tr\{ABCD\}=0$ implies that $tr\{BD\}=0$.

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