A matrix to the power of zero gives identity matrix even if it doesn't have an inverse?

If one matrix whose determinant is equal to 0 which means it doesn't have an inverse. Then how is possible to find the value of the matrix to the power of 0 equal to identity matrix when multiplying the original matrix with something undefined?

Is it a math fluke, or I am missing some important information?

This is a good question.

The reason we define $A^0 = I$ is so that the identity $$ A^{m+n} = A^mA^n $$ holds whenever $m$ and $n$ are nonnegative integers. Then we can evaluate $p(A)$ for any polynomial $p$, and sometimes even compute with power series to get things like $e^A$.

This all makes sense and is often useful whether or not $A$ is invertible. When it is, we extend the definition so that $A^{-n} = (A^{-1})^n$.