is there a nice way to calculate the inverse of the following matrix symbolically?
$A\in\mathbb{R}^{n\times n+1}$ and $b\in\mathbb{R}^{1\times n+1}$ so that $B=\left(\begin{array}{c}A\\b \end{array}\right)\in\mathbb{R}^{n+1\times n+1}$ are given. Is there a nice way to express the inverse of B , inv(B) or is that not possible?
$$\begin{bmatrix} \mathrm A\,\,\\ \mathrm b^{\top}\end{bmatrix} \begin{bmatrix} \mathrm X & \mathrm y\end{bmatrix} = \begin{bmatrix} \mathrm I_n & 0_n\\ 0_n^{\top} & 1\end{bmatrix}$$
produces $4$ linear matrix equations. Vectorizing, we obtain the following linear system
$$\left( \begin{bmatrix} \mathrm I_n & 0_n\\ 0_n^{\top} & 1\end{bmatrix} \otimes \begin{bmatrix} \mathrm A\,\,\\ \mathrm b^{\top}\end{bmatrix} \right) \begin{bmatrix} \mbox{vec} (\mathrm X)\\ \mathrm y\end{bmatrix} = \begin{bmatrix} \mbox{vec} (\mathrm I_n)\\ 0_n\\ 0_n\\ 1\end{bmatrix}$$