pronouns - Whom did she invite or Who did she invite?

Whilst in most cases, I'm clear using who or whom but this is one of the sentences bothering me!

Whom/who did she invite?

Whom did she invite looks very very natural to me. The general rule (this will work in most of the cases, no exceptions please!) I remember is... be yourself in that situation, and if 'I' fits, it'll be 'who' and if 'me' fits, it'll be 'whom'.

Some examples where this rule works as a breakthrough...

Whom this matter is concerned to? - The matter is concerned to me and not I so whom!
Our madam wondered who broke the glass - I broke the glass and not me so who!

That way, She invited me and not I so it should be - whom did she invite, shouldn't it? Also, if you take it other way - She invited him and not he.

Answer

"Whom did she invite?" is grammatically correct, for the reasons you have explained. As Stan and Hunter have explained, this rule is often broken, especially in conversation.

Personally, I cleverly evade the issue in conversation by saying "Who'd she invite?" If a "grammar fiend" (thanks Hunter) attempted to correct me, I would explain to said grammar fiend that "who'd" is a contraction of either "whom did" or "who did" depending on context, and invite him to prove otherwise. I would then walk away and talk to someone else. :)

One more thing. Rather than as you have it, correct is "Whom does this matter concern? It concerns me." Whether a verb can function as transitive or not can be pretty arbitrary in English. If you look up "concern" here: http://www.merriam-webster.com/dictionary/concern you will see that this one is transitive.