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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?

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2 Answers 2

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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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    $\begingroup$ beat me by 5s... $\endgroup$
    – acl
    Commented Jan 26, 2012 at 16:25
  • $\begingroup$ He was also asking for the inverse - could add this for completeness. $\endgroup$
    – Vitaliy Kaurov
    Commented Jan 26, 2012 at 16:31
  • $\begingroup$ @VitaliyKaurov, done! $\endgroup$
    – rcollyer
    Commented Jan 26, 2012 at 16:36
  • $\begingroup$ @acl, it's a testament to how much we've been waiting for these types of questions when 3 of the top users jump on it within seconds of each other. $\endgroup$
    – rcollyer
    Commented Jan 26, 2012 at 16:38
  • $\begingroup$ This point about {t}\[Transpose] being necessary is important and subtle: Mathematica (to its credit) does NOT finesse the fact that vectors must be either 1 x n or n x 1 matrices, that is, explicitly either row or column vectors. Pretty much everything else (including my beloved Golub & VanLoan) does (tho G&VL have the grace at least to say they do). I find the conceptual hygiene forced by Mathematica to be refreshing and helpful. $\endgroup$
    – Reb.Cabin
    Commented Jul 27, 2014 at 17:32
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The keyboard commands Ctrl+Enter, Ctrl+, and Tab can be used to enter this format.

You can also use the menu Insert > Table/Matrix to create a table of specified size with placeholders.

See Entering Tables and Matrices.

Depending on the meaning of the question, this may have some bearing:

Mathematica graphics

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    $\begingroup$ beat you to it. :P $\endgroup$
    – rcollyer
    Commented Jan 26, 2012 at 16:23
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    $\begingroup$ LOL at the pity vote. $\endgroup$
    – Mr.Wizard
    Commented Jan 26, 2012 at 16:27
  • $\begingroup$ I meant, I beat you to posting the "correct" answer. Although, I posted the result he wanted, and you posted the entry method. Win for both of us. $\endgroup$
    – rcollyer
    Commented Jan 26, 2012 at 16:28
  • $\begingroup$ Not a pity vote. If you don't post this, I would have edited it into @rcollyer's answer. It's much more convenient to enter matrices like this (see my screenshot). $\endgroup$
    – Szabolcs
    Commented Jan 26, 2012 at 16:29
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    $\begingroup$ @rcollyer Recently I saw someone give input to NDSolve that way, and I really liked it --> i.sstatic.net/bgWJ3.png $\endgroup$
    – Szabolcs
    Commented Jan 26, 2012 at 16:47

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