Final answer:
The claim that any probability matrix of full rank can be decomposed into a sum of matrices of rank 1 is false, as the unique properties of probability matrices can prevent such a decomposition.
Step-by-step explanation:
The statement that every probability matrix P of rank m, which is a full-rank matrix, can be decomposed into a sum of m probability matrices of rank 1 is false. While a matrix can often be decomposed into a sum of matrices of rank 1, known as a rank-1 decomposition, this result does not universally apply to probability matrices. In probability theory, the rank of a probability matrix refers to the maximal number of linearly independent rows (or columns) in the matrix. These matrices must also satisfy certain conditions, such as all entries being non-negative and each row summing to 1, to qualify as probability matrices. Thus, the specific properties of probability matrices may prevent them from being decomposable into a sum of rank-1 probability matrices.