Let $X$ be an n×n real or complex matrix. The exponential of $X$, denoted by $e^X$, is the n×n matrix given by the power series $e^X =\sum_{k=0}^{\infty} X^k/k!$ where $X^{0}$ is defined to be the identity matrix $I$ with the same dimensions as $X$.
If $X$ is zero matrix then $e^X$=$I_{n \times n}$ or $0_{n \times n}$?
On
$e^X=I_n$ by definition if $X=0_n$. See the wiki https://en.m.wikipedia.org/wiki/Matrix_exponential
It is the identity matrix. The zero matrix to the zeroth power is the identity matrix, $0_{n,n}^0 = I_{n,n}$, for the same reason that $0^0 = 1$ in the numbers. And all the higher terms are zero since $0_{n,n}^k = 0_{n,n}$ for all $k \ge 1$. In symbols: $$e^{0_{n,n}} = 0_{n,n}^0 + 0_{n,n}^1 + \frac{1}{2} 0_{n,n}^2 + \dots = I_{n,n}.$$