# What is zero power of a non-square matrix?

I'm told that zero power of square matrix is an identity matrix of appropriate size. How it is with a non-square matrix?

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Exponentiation of (square) matrices is a recursive function, which we define so that $A^0 =I$ and $A^1 = AA^0 = A^0A$, $A^2 = AA^1= A^1A$, etc.
Exponentiation of non-square matrices is not well-defined, for the simple reason that $A^2 =AA$ (and higher products of $A$ with itself) is not a valid matrix product. So, the short answer is that because we don't define exponentiation at all for non-square matrices, even $A^0$ and $A^1$ are undefined in the non-square case.
Suppose you insisted on having a definition of $A^0$ and $A^1$ for m×n matrices for mn, so that $A^1=A$ and also $A^0$ was some matrix which gave you back $A$ after left- or right-multiplication by $A$. But that's absurd, because whatever you tried to define $A^0$ as, only one of those products would be defined. So we don't really have any good, unique choice for what $A^0$ should be that would also give us a reasonable definition of $A^1$ that we would expect. Lacking any good way to define $A^0$, we leave both $A^0$ and $A^1$ undefined.