Is there a shortcut to invert this 3x3 matrix?

Question

Is there an easy way to invert the matrix on right-hand-side? Like is there some quick formula or observation we can make or do I have to do it manually all the way? Please, dont be harsh on me since I am not good at linear algebra.

EDIT: Result

Answer 1

Generally if $A$ is a block matrix, that is a matrix of the form

$$\begin{pmatrix} A_1 \\ & A_2 \\ & & \ddots \\ & & & A_n \end{pmatrix}$$

for square matrices $A_i$, and zeroes in all the blank spots, and if all the matrices $A_i$ are invertible, then $A$ is also invertible, with inverse

$$\begin{pmatrix} A_1^{-1} \\ & A_2^{-1} \\ & & \ddots \\ & & & A_n^{-1} \end{pmatrix}.$$

You should convince yourself of this with some small examples. Anyway, do you know how to quickly inverse a $2$ by $2$ matrix? By the principle I mentioned, you can immediately invert your $3$ by $3$ matrix, which is a block matrix consisting of the $2$ by $2$ matrix

$$\begin{pmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{pmatrix}$$

and the $1$ by $1$ matrix $(1)$.