Example of a 3x3 Matrix with no inverse, where the row totals add to 1, and no duplicate rows

Question

Is it possible to have a 3x3 matrix that has no inverse, where the row totals add up to 1, and there are no duplicate rows? And, can this be extended to an nxn matrix? I am inclined to say no, because in order for a matrix to have no inverse, the rows must be linearly dependent? Every example I can think of, yields a matrix that has row totals that add up to 1, but there are duplicate rows.

Answer 1

HINT

In $\mathbb{R}^2$, the set of vectors $(0.1,0.9), (0.2,0.8), (0.3,0.7)$ is certainly linearly dependent even though each vector's coordinates add up to $1$. Can you think of them as vectors in $\mathbb{R}^3$?