Sufficient conditions for total positivity, compounds, and Dodgson condensation

A n-by-n matrix is called totally positive ($TP$) if all its minors are positive and TPₖ if all of its k-by-k submatrices are $TP$. For an arbitrary totally positive matrix or TPₖ matrix, we investigate if the rth compound ($1<r<n$) is in turn $TP$ or TPₖ, and demonstrate a strong negative resolution in general. Focus is then shifted to Dodgson's algorithm for calculating the determinant of a generic matrix, and we analyze whether the associated condensed matrices are possibly totally positive or TPₖ. We also show that all condensed matrices associated with a $TP$ Hankel matrix are $TP$.