Inverse of Matrix Exponential

Theorem
Let $\mathbf A$ be a square matrix.

Let $t \in \R$ be a real number.

Let $e^{\mathbf A t}$ denote the matrix exponential of $\mathbf A$.

Then:
 * $\paren {e^{\mathbf A t} }^{-1} = e^{-\mathbf A t}$

where $\paren {e^{\mathbf A t} }^{-1}$ denotes the inverse of $e^{\mathbf A t}$.

Proof
where:
 * $\mathbf 0$ denotes the zero matrix of the appropriate order
 * $\mathbf I$ denotes the identity matrix of the appropriate order.

Similarly:
 * $e^{-\mathbf A t} e^{\mathbf A t} = \mathbf I$

Hence the result by definition of inverse matrix.