WHO KILLED ALEXANDER THE GREATS'S HALF BROTHER ARRHADEUS, SON OF
CLEOPATRA,  AND WHO IS RESPONSIBLE FOR HAVING HIM KILLED?

You may find some additional information when searching for Arridaeus
or Arrhidaeus. However, his mother's name wasn't Cleopatra.

.............of course Cleopatra the famous one, lived 300 years after
Alex the boy wonder died. Her son was a nobody carpenter or something
similar and died a nobody as well.

Arridaeus was the son of "a Thessalian woman named Philinna":
http://www.livius.org/phi-php/philip/arridaeus.htm

There are many webpages mentioning Philinna, such as e.g.
http://www.1stmuse.com/alex3/argeaidos.html

You may also find these webpages interesting:
http://www.archaeonia.com/history/hellenistic/heritage.htm
http://www.royalty.nu/Europe/Balkan/Alexander.html

Our best source for details of the death of Philip Arridaeus (on Dec.
25, 317 BCE) is the Library of World History by Diodorus, written in
the middle of 1st century BCE.

You can read a translation of the relevant passage here:

[The Diadochi: The death of Philip Arridaeus]
http://www.livius.org/di-dn/diadochi/diadochi_t15.html

Olympias, the mother of Alexander the Great, "ordered certain
Thracians to stab Philip to death, who had been king for six years and
four months"

Diodorus of Sicily used many sources in preparing his immense forty
volume history of the world from mythological beginnings to the early
successes of Julius Caesar.  Sixteen of these books survive in
complete form, and fragments of others through quotation by later
authors.  See here for a description of his life work and what is
known about the sources available to him:

[Diodorus of Sicily]
http://www.livius.org/di-dn/diodorus/siculus.html

regards, mathtalk-ga