Before revision:  and B = (b_{ij}) , denoted by \mathbb{I is the collection of all matrices C such that c_{ij=t_{ij}a_{ij} +(1-t_{ij})b_{ij}for all i,j, where t_{ij} \in } 0,1^{m \times n} such that each c_{ij} is between a_{ij} and b_{ij}} . Using the sign non-reversal property, we identify a finite subset of \mathbb{I}(A,B) that detects the total positivity of all of \mathbb{I}(A,B) . This provides a test for  an entire class of matrices simultaneously to be totally positive.
After revision:  and B = (b_{ij}) , denoted by \mathbb{I is the collection of all matrices C such that c_{ij=t_{ij}a_{ij} +(1-t_{ij})b_{ij}for all i,j, where t_{ij} \in } 0,1^{m \times n} such that each c_{ij} is between a_{ij} and b_{ij}} . Using the sign non-reversal property, we identify a finite subset of \mathbb{I}(A,B) that detects the total positivity of all of \mathbb{I}(A,B) for arbitrary k \geq 1. In particular, this provides a test for an entire class of matrices simultaneously to be totally positive.