91Ó°ÊÓ

Given that A is a transition matrix and B is a positive transition matrix, where A andB are of the same size.

By definition of transition matrix, every column of Ais a distribution vector and their sum is 1.

Thus, A cannot have all-zero columns(since sum should be 1), but it can have an all-zero row.

The ${(i,j)}^{th}$entry of the matrix AB is the dot product of the$i^{th}$ row of A and the$j^{th}$ column of B.

Now, the row of A could be all zero so the dot product could be zero.

Therefore, the matrix AB is not necessarily a positive transition matrix.

Now, the${(i,j)}^{th}$ entry of the matrix BA is the dot product of the $i^{th}$row of B and the$j^{th}$ column of A. As B is a positive transition matrix, every element of the row vector is positive and at least one element of the column vector is positive (since it is a distribution vector).

Thus, the dot product is positive.

Hence, BA is a positive transition matrix.