Linear algebra- Find inverse for block matrix

Question

I have a 3*3 matrix of the form $\begin{bmatrix} a & 0 \\ 0 & D \\ \end{bmatrix}$, where a is a constant and D is a 2*2 matrix. How can I find the inverse of the matrix ?

Also, for a more general case with $\begin{bmatrix} A & 0 \\ 0 & D \\ \end{bmatrix}$, A and D both matrices. How can I express the inverse?

Thnaks!

Answer 1

For the inverse matrix, replace $a$ by $\frac{1}{a}$ and the matrix $D$ by its inverse

In the general case you also replace the matrix $A$ by its inverse