I just read in one of the questions answered by @MikeSpivey that the following table is provided in Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144.
Primal Optimal Solution Dual Optimal Solution (a) Multiple implies Degenerate (b) Unique and nondegenerate implies Unique and nondegenerate (c) Multiple and nondegenerate implies Unique and degenerate (d) Unique and degenerate implies Multiple
I wonder if the information provided in this table is provided based on the assumption that matrix A has full rank.