I don't know the history of the two phrases, but today they have different meanings ("of course" cannot be used as an abbreviation for "as a matter of course").

"Of course" introduces a statement that the reader and the writer are both likely to understand and agree on.

"As a matter of course" means "routinely". For example:
When you enter the US, your passport is inspected as a matter of course.

http://www.meriam-webster.com/dictionary/course

@ GeorgeA, thats what I thought until I read the link now`I`m confused again(its not hard to confuse me)

Leo übersetzt "Selbstverständlichkeit" mit "as a matter of course".
"Of course" bedeutet "selbsverständlich".
Also:
Of course, we are going to the beach on Tuesday
(selbstverständlich gehen wir...)
As a matter of course, we are going to the beach on Tuesday
(es ist eine Selbstverständlichkeit, daß wir ... gehen)

Der Zusammenhang scheint mir irgendwie gegeben zu sein.

Aber dein zweiter Satz ist erstens nicht ganz richtig, ich würde da "As a matter of course we go to the beach on tuesdays" sagen, und zweitens drückt "as a matter of course" eben aus, dass die Selbstverständlichkeit, mit der etwas getan wird, aus einer Routine heraus kommt. "Of course" kann dagegen durchaus auch bei einem einmaligen Vorgang verwendet werden. Wie schon GeorgeA weiß auch ich nicht, ob dieser Unterschied schon immer bestand oder ob sich "of course" tatsächlich aus "as a matter of course" entwickelt hat.

They are related, as they both use the word "course", which means "the way things normally go", among other things. So IMHO they are related in the same way that "seinen Lauf nehmen" and "der Lauf der Welt" are related, rather than one being a short form of the other.

course noun ... 4 the normal progress of something. ... in due course at the appropriate or expected time. a matter of course a natural or expected action or result. of course 1 as expected. (Chambers)

OUP:

in the ordinary, normal, etc. course of events, things, etc. as things usually happen normally: In the normal course of things we would not treat her disappearance as suspicious.

run / take its course to develop in the usual way and come to the usual end: When her tears had run their course, she felt calmer and more in control. With minor ailments the best thing is often to let nature take its course.—

& also (OUP): (as) a matter of course (as) the usual and correct thing to do: We always check people’s addresses as a matter of course.

So "We are going to the beach on Tuesday as a matter of course" means that you are going there without questioning why you should not; you have a very clear reason for doing something. For example, if you are a Royalist and you heard the Queen will be visiting Clacton next week, you might say this.

"Of course we are going to the beach on Tuesday" means only that it is obvious that you are going there. For example, if your son says "Dad, please can we go? Please? Please?" then you can kindly say "Of course we are going to the beach, son!" (i.e.: you know I'm such a nice dad!). Or if your secretary is a bit stupid and you have told her you are going three times, and she still asks if you are going, you can snap: "OF COURSE we are going to the beach, woman!"

Murphy's Law of Evolving Systems:

Left to themselves, things go from bad to worse.

Wenn schon zitieren, dann aber richtig:

Left to themselves, things tend to go from bad to worse.