Your queries:
select A from table_A where A like 'A%'
and
select A from table_A where A like 'A%%'
^ optimizer will remove second redundant %
are equivalent, the optimizer will remove the second % in the second query
just like it would remove the 1=1 from:
select A from table_A where A like 'A%%' and 1=1
However, this query is very different:
select A from table_A where A like '%A%'
The when using 'A%' it will use the index to find everything starting with an A, like a person using a phone book would quickly look for the start of a name. However when using '%A%' it will scan the entire table looking for anything containing an A, thus slower and no index usage. Like if you had to find every name in the phone book that contained an A, that would take a while!