Inverse of block matrix which satisfies some ODEs

Question

Let $A(t) \in \mathbb{R}^{2n\times 2n}$, $B(t) \in \mathbb{R}^{n\times n}$ be smooth and bounded. Suppose $\Phi(t) \in \mathbb{R}^{2n\times 2n}$ solves $d\Phi(t)=A(t)\Phi(t)dt$ and $\Phi(0)=I_{2n}$ for $t\in [0,T]$, then the inverse also exists and solves $d[\Phi(t)]^{-1}=-[\Phi(t)]^{-1}A(t) dt$ and $[\Phi(0)]^{-1}=I_{2n}$ for $t\in [0,T]$. Given that $$ \begin{pmatrix} B(t) & I_n \end{pmatrix} \Phi(t) \begin{pmatrix} \mathbf{0}\\I_n \end{pmatrix} $$ has an inverse for any $t\in [0,T]$ and the inverse is also bounded. I would like to show that
$$ \begin{pmatrix} B(t) & I_n \end{pmatrix} \big[\Phi(t)\big]^{-1} \begin{pmatrix} \mathbf{0}\\I_n \end{pmatrix} $$ has an inverse for any $t\in [0,T]$ and the inverse is also bounded. Is there any method?