The purpose of the determining flow is to be able to distinguish 1) the flow that makes an activity occur 2) the flows that occurs as a consequence of the activity occurring.
It will allow us to distinguish two otherwise identical activities, such as sheep farming for wool and sheep farming for meat
If we do not have a place to indicate this, we would loose information that is essential for the linking of the flows between activities. The identification of determining flows is based on empirical observation and algorithms, i.e. it is not a value choice or something that will depend on the application of the data in different contexts.
It does not have anything to do with square matrices as such, but in a square matrix there is often the convention to place the determining flow on the diagonal.