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## It is defined that for any matrix A, $A^{0}$ is defined

 Quote by micromass Sure. I doubt you'll find much uses. Defining $0^0=1$ is really useful and it simplifies a lot of formulas. It is obvious that it is much more useful than defining it to be 2.
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.

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When somebody writes down the binomial theorem, then I have never seen him say explicitely that we assume $0^0=1$.
If $\kappa$ and $\kappa^\prime$ are cardinal numbers and if X is a set with cardinality $\kappa$ and Y has cardinality $\kappa^\prime$. Then we say by definition that $X^Y$ has cardinality $\kappa^{\kappa^\prime}$.
If you take in that definition $\kappa=\kappa^\prime=0$. Then by definition, the cardinal $0^0$ is the size of the set $\emptyset^\emptyset$. This size is one.
So at least for cardinal numbers, we do make the explicit and general definition $0^0=1$.
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