linear algebra - How to enter matrices in block matrix format?

Example: I have a matrix $R = \left( \begin{array}{cc} A & \mathbf{t} \\ 0 & 1 \end{array} \right) $ where $A$ is 3-by-3 and $\mathbf{t}$ is 3 by 1. Or in Mathematica

 A={{1,0,0},{0,0,1},{0,-1,0}};
 t={1,1,1}

I would like to be able to use a form of block matrix notation / entry and subsequently find the inverse of R.

Question: Is this possible?

Answer

You're looking for ArrayFlatten. For your example matrices,

 R = ArrayFlatten[ {{A, {t}\[Transpose]},{0, 1}} ]
 (*
 => {{1, 0, 0, 1}, {0, 0, 1, 1}, {0, -1, 0, 1}, {0, 0, 0, 1}}
 *)

The construct {t}\[Transpose] is necessary for ArrayFlatten to treat t as a column matrix.

Mathematica graphics

Then to find $\boldsymbol{R}^{-1}$, you run

Inverse[R]
(* 
=> {{1, 0, 0, -1}, {0, 0, -1, 1}, {0, 1, 0, -1}, {0, 0, 0, 1}}
*)

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