Is the $0\times 0$ matrix (zero-times-zero matrix) a well-defined concept? Is the $0\times 0$ matrix a well-defined concept, and if yes, what can be said about it?
Intuitively it should be a well-defined concept, since we have the zero vector space, and every linear mapping between vector spaces (such as the zero mapping from the zero vector space to itself) can be represented by a matrix.
The determinant of a $0\times 0$ matrix should be $1$, intuitively, just as the zero-dimensional volume of a point is $1$.
If the $0\times 0$ matrix is a bad concept or ill-defined concept, can you explain where to find the catch?
 A: Well, one could say even more about the $0$ x $0$ matrix: Yes, it operates on the zero vector space (which contains only one single element = $0$). So, it maps $0$ to $0$ since there is no other possible image. Hence, it must be the identity mapping on the zero space, and therefore, it is its own inverse.
A: Suppose $A$ is a linear operator on a finite-dimensional $K$-vector space $V$. Say $\dim_K V = n$. It induces a natural linear map $\wedge^n A$ on the exterior power $\wedge^n V$. The latter is a $1$-dimensional $K$-vector space, so that $\wedge^n A$ is multiplication by a scalar. This scalar is the determinant of $A$.
(This is the definition of determinant you'll find in Bourbaki.)
When $n = 0$, we have a natural identification $\wedge^0 V \cong K$, and the induced map $\wedge^0 A$ is (by construction) the identity map. I.e. multiplication by $1$. Thus $\det A = 1$.
A: It breaks one important rule: for other square matrices of rank $n$ we can represent them isomorphically as matrices of dimension $n\cdot m$.
For instance, the matrix $\left(
\begin{array}{cc}
 a & b \\
 c & d \\
\end{array}
\right)$ can be isomorphically represented as $\left(
\begin{array}{cccc}
 a & b & 0 & 0 \\
 c & d & 0 & 0 \\
 0 & 0 & a & b \\
 0 & 0 & c & d \\
\end{array}
\right)$ or as $\left(
\begin{array}{cccc}
 a & 0 & b & 0 \\
 0 & a & 0 & b \\
 c & 0 & d & 0 \\
 0 & c & 0 & d \\
\end{array}
\right)$. But we cannot represent a $0\times0$ matrix in higher dimension.
