If $AB$ is diagonally dominant, is $ADB$ diagonally dominant for positive diagonal $D$? Let $A$ and $B$ be non-square real matrices such that $AB$ is diagonally dominant. Let $D$ be a positive diagonal real matrix. Is $ADB$ diagonally dominant?

Another counterexample, with smaller numbers . . .

Let $A,D,B$ be given by $$ A = \pmatrix { 1 & 1 & 1 \cr 1 & 1 & 2 } ,\;\; D = \pmatrix { 1 & 0 & 0 \cr 0 & 1 & 0\cr 0 & 0 & 2 } ,\;\; B = \pmatrix { 2 & 1 \cr 3 & 1 \cr 1 & 3 } $$ Then we have $$AB = \pmatrix { 6 & 5 \cr 7 & 8 } \\\\[20pt] ADB = \pmatrix { 7 & 8 \cr 9 & 14 } $$ so $AB$ is diagonally dominant, but $ADB$ is not diagonally dominant.