Holte introduced a [math] matrix [math] as a transition matrix related to the carries obtained when summing [math] numbers base [math] . Since then Diaconis and Fulman have further studied this matrix proving it to also be a transition matrix related to the process of [math] -riffle shuffling [math] cards. They also conjectured that the matrix [math] is totally nonnegative. In this paper, the matrix [math] is written as a product of a totally nonnegative matrix and an upper triangular matrix. The positivity of the leading principal minors for general [math] and [math] is proven as well as the nonnegativity of minors composed from initial columns and arbitrary rows.