Let $\mathbb{M}(n,\mathbb{F})$ be the set of all $n \times n$ matrices over a field $\mathbb{F}$. I now take the subset comprising all matrices that are matrix exponentials of some other matrices
$$A=\exp (B)$$
The properties of the matrix exponential are such that all of these matrices are invertible
$$A A^{-1} = \exp (B) \exp(-B) = \exp(B-B)= \exp (0) = I$$
Is the subset of all invertible matrices the same as the subset of all matrices that can be expressed as exponentials?