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Suppose $A$ and $B$ are real symmetric, positive definite, strictly diagonally dominant matrices with positive diagonal and nonpositive off-diagonal elements (i.e., strictly diagonally dominant Stieltjes matrices).

Let $C^{-1}=A^{-1}+B^{-1}$.

Is $C$ also a strictly diagonally dominant Stieltjes matrix?

Note: The result holds for a 2-by-2 matrix and I did not find any counter-examples for slightly larger sized matrices, however I wasn't able to figure out how to prove the result for a general $n$-by-$n$ matrix.

I would be grateful for any comments and guidance. Thank you so much in advance!

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No. Random counterexample: $$ A=\pmatrix{ 110&-88&-10\\ -88&160&-69\\ -10&-69&129}, \ B=\pmatrix{ 118&-13&-82\\ -13& 66&-52\\ -82&-52&140}. $$ $$ C=(A^{-1}+B^{-1})^{-1}=\pmatrix{ 48.416&-19.298&-20.637\\ -19.298&45.099&-26.202\\ -20.637&-26.202&58.454}. $$ The second row of $C$ is not diagonally dominant.

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    Thank you so much for your help! – 2017-01-22
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    If I may, I would like to ask a follow up question to @user1551 or anyone else who would be interested. If one removes the strict-diagonally dominant condition, would the statement hold then? In other words, supposing $A$ and $B$ are real symmetric, positive definite matrices with positive diagonal and nonpositive off-diagonal elements (i.e., Stieltjes matrices). Would $C$, as defined earlier, then hold these properties as well? Thank you in advance! – 2017-01-23
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    @opre No. Clearly $C$ is positive definite (and hence it has a positive diagonal), but off-diagonal entries of $C$ can be positive too. It's easy to generate a 3x3 counterexample by computer. E.g. when $A=\pmatrix{10&-7&-2\\ -7&10&-3\\ -2&-3&10}$ and $B=\pmatrix{15&-2&-2\\ -2&15&-9\\ -2&-9&9}$, the top right entry of $C$ is about 0.053545. If you have further questions, the following paper may be useful: C.R. Johnson, [*Inverse M-matrices*](http://www.sciencedirect.com/science/article/pii/0024379582902385), LAA 47:195-216(1982). – 2017-01-23
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    Thanks so much again @user1551, also for the reference to the article. I was aware of a few papers on this area but wanted to see if there are any stronger results particularly for a symmetric matrix as much of the treatment and results are for more general asymmetric matrices. Thank you. – 2017-01-23