## Conclusion

This summer I worked on a mathematical proof. I, working with Professor Johnson, proved that a partial TP (TN) matrix with a border pattern is TP (TN) completable. Though TP and TN are very similar properties, to show their completability require two different proofs. In TN case, we developed a Northwest principle and a (m,n) entry argument to show competability. In TP case, penultimate line insertion was used to show TP competability. The insertion is only possible with a previous theorem proved in S.M. Fallat et al. (2000).

## penultimate line insertion

In this post, I will describe the penultimate line insertion method derived from a paper by S.M. Fallat, C.R. Johnson and Ronald L. Smith. This step is important because it will eventually help us show that a partial TP matrix with a border pattern is TP completable.

## Border pattern TN completion

In this blog, I will show that a m-by-n partial Tn matrix with a border pattern can be completed to a TN matrix. I will use the Northwest principle and Lemma 1 from the previous blog.