The pullback of 1-forms on $\textrm{Gl}(n;\mathbb R)$ by an exponential map.

Question

This is from P53 of Differential Geometry by Taubes. Fix some $q\in \textrm M (n;\mathbb R)$. Set the 1-form $\omega_q$ on $\textrm{Gl}(n;\mathbb R)$ $$\omega_q|_m=\textrm{tr}(qm^{-1}\textrm d m),$$ where $\textrm d m$ denotes the matrix whose $(i,j)$-th entry is $\textrm dm_{i,j}$. The exponential map $e_\iota$ is $$e_\iota: \textrm M (n;\mathbb R) \to \textrm{Gl}(n;\mathbb R),m\mapsto e^m.$$ Then Taubes asserts that the pullback of $\omega_q$ by $e_\iota$ is $$e_\iota^*\omega_q=\int^1_0 \mathrm{d}s~ \text{tr}(e^{-sa}qe^{sa}\textrm da).$$ That's where I stuck. Why the pullback is like that?