matrix product with trace zero

Question

$D$ is a positive definite matrix, $A$ and $B$ are both positive semidefinite matrices, $c$ is a postive integer. I want to know whether $trace\{(A+B+cI)^{-1}ABD\}=0$ implies that $AB=0$?

Answer 1

No. Let $$A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\!, \quad B = \begin{bmatrix} 1 \\ & 0 \end{bmatrix}\!, \quad D = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}\!, \quad c = 1.$$ Then $$(A + B + cI)^{-1} A B D = \frac{1}{5}\begin{bmatrix} 2 & -1 \\ 4 & -2 \end{bmatrix}\!,$$ so $\mathop{\rm tr}((A + B + cI)^{-1} A B D) = 0$, but $$AB = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}.$$

Acknowledgment: Example fixed by Sebastien B's comments.