It is defined that for any matrix A, [itex] A^{0} [/itex] is defined

arildno

And in those formulas, you evidently provide the EXPLICIT definition of 0^0 being..1, thus making it a particular case valid for that formula, rather than any general result or definition.

micromass

When somebody writes down the binomial theorem, then I have never seen him say explicitely that we assume [itex]0^0=1[/itex].

But in general, you certainly do make a general definition.
If [itex]\kappa[/itex] and [itex]\kappa^\prime[/itex] are cardinal numbers and if X is a set with cardinality [itex]\kappa[/itex] and Y has cardinality [itex]\kappa^\prime[/itex]. Then we say by definition that [itex]X^Y[/itex] has cardinality [itex]\kappa^{\kappa^\prime}[/itex].

If you take in that definition [itex]\kappa=\kappa^\prime=0[/itex]. Then by definition, the cardinal [itex]0^0[/itex] is the size of the set [itex]\emptyset^\emptyset[/itex]. This size is one.
So at least for cardinal numbers, we do make the explicit and general definition [itex]0^0=1[/itex].