Invalid Argument vs. Contradiction? I'm confused about the differences between valid/invalid arguments and contradictions/tautologies.
An argument is valid when the statement "all of it's hypothesis are true implies the conclusion" is a tautology.  
When we show an argument is invalid we are showing the above quoted implication-statement is false in at least one case.  Right?
Do we ever need to prove that an argument is equivalent to a contradiction?  I am trying to think of an example where this would be the case, or even could be the case, but...

An argument that from $A$ derives $B$ is valid if and only if $A \Rightarrow B$ can be derived from the starting axioms. Derivability should be distinguished from truth value because the assignment of truth values to statements according to derivability will of course depend on the rules in our formal system, and may have nothing to do with truth in the actual world we live in, unless the real world follows exactly the same rules as our system. Also, in some formal systems there are statements such that neither it nor its negation can be derived. However if we make the assumption that our formal system has no such statements, and is powerful enough for us to describe and manipulate its statements and derivations completely within it, then it makes sense for us to assign derivable statements the value of true and non-derivable well-formed statements the value of false, and then we can indeed say that an argument that from $A$ derives $B$ is valid if and only if $x$ is true, where $x$ is the statement ( $A \Rightarrow B$ is true for all possible values for all the variables ). Likewise an argument that from $A$ derives $B$ is invalid if and only if $x$ is false. In such systems, every invalid argument corresponds to a false statement with no free variables, which is a contradiction.
The reason why you could not think of an example where you want to prove that an argument is a contradiction is precisely because of the quantification over the variables as described above. A valid argument must be true for all possible bindings of free variables, whereas an invalid argument must just be false for at least one possible binding, which is not necessarily contradictory unless it is further 'wrapped up' in universal quantifiers to bind all its free variables. In other words, it may not be a contradiction because it may be true for some bindings, but it becomes a contradiction if you encapsulate it into a statement saying that it is true for all bindings.
