Find inverse of specific matrix

Question

So, I have to find an inverse of this matrix: $$\begin{pmatrix} A & B\\ 0 & C \end{pmatrix}$$ where, $A\in M_m(\mathbb{R})$,$B\in M_{mn}(\mathbb{R})$, $C\in M_n(\mathbb{R})$ and $A$ and $C$ are both invertible.

I've tried $$\begin{pmatrix} A^{-1} & 0\\ 0 & C^{-1} \end{pmatrix}$$

But it doesn't work

you mean $A$ and $C$ are invertible? an $m\times n$ matrix can't be invertible, unless $n=m$. — Yanko, Aug 17 '19 at 21:16

score 3 · Accepted Answer · answered Aug 17 '19 at 21:17

3

Try $$\begin{pmatrix} A^{-1} & -A^{-1}BC^{-1}\\ 0 & C^{-1} \end{pmatrix}$$

To come up with that I simply calculated the inverse of $$\begin{pmatrix} a & b\\ 0 & c \end{pmatrix}$$ where $a,b,c\in\mathbb{R}$ in the usual way.

answered Aug 17 '19 at 21:17

Yanko

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Find inverse of specific matrix

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