Is a regular stochastic matrix definitely nonsingular?

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Is a regular stochastic matrix definitely nonsingular (invertible)? How to prove it ?

It says here that 'For a regular matrix always an inverse matrix exists' http://www.vias.org/tmdatanaleng/hl_regularmatrix.html

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Certainly not always. Consider the all-one matrix normalised by its size.

By the way, what you call a regular matrix is usually known as a primitive matrix.