How to write $P^{-1}$ in terms of $Q^{-1}$?

Question

I have the following $n \times n$ matrix

$$P = \begin{bmatrix} a & \begin{matrix} 0_{n-1}^{\top} \end{matrix} \\ \begin{matrix} 0_{n-1} \end{matrix} & \text{ $Q$} \end{bmatrix}$$

where $a \in \mathbb R$ and $Q$ is an $(n-1) \times (n-1)$ matrix. Is it possible to find $P^{-1}$ in terms of $Q^{-1}$?

I am unable to come up with a technique to find $P^{-1}$ in terms of $Q^{-1}$. I will be grateful for some help.

Answer 1

This is a block diagonal matrix. To invert it, simply invert each diagonal block, i.e. $$P^{-1}=\begin{bmatrix} \tfrac{1}{a} & \begin{matrix} 0 & 0 & \dots & 0 \end{matrix} \\ \begin{matrix} 0 \\ 0 \\ \vdots \\[1ex] 0 \end{matrix} & \text{ $Q^{-1}$} \end{bmatrix}.$$

Answer 2

How about: $$ P^{-1}=\begin{bmatrix} 1/a & \begin{matrix} 0 & 0 & \dots & 0 \end{matrix} \\ \begin{matrix} 0 \\ 0 \\ \vdots \\[1ex] 0 \end{matrix} & \text{ $Q^{-1}$} \end{bmatrix} $$ This is what you get from Gaussian elimination $[P\mid I]\to\dots$