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Are these kind of matrices generally known in mathematics? Do they have a name?

$ \left[\begin{array}{rrr} A & B \\ B & A \\ \end{array}\right] $

$ \left[\begin{array}{rrr} A & B & C \\ C & A & B \\ B & C & A \\ \end{array}\right] $

$ \left[\begin{array}{rrr} A & B & C & D \\ D & A & B & C \\ C & D & A & B \\ B & C & D & A \\ \end{array}\right] $

$ \left[\begin{array}{rrr} A & B & C & D & E \\ E & A & B & C & D \\ D & E & A & B & C \\ C & D & E & A & B \\ B & C & D & E & A \\ \end{array}\right] $

The main thing is that each letter will be in the same columnn/row just once. I'm trying to do some combination calculations with big matrices following this pattern, so knowing effective ways to generate and compute these would help.

(The pattern here is that the next row is made by shifting the previous row to one right.)

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    If the main idea is that each letter appears in each row and each column exactly once, then there are many examples of that, that do not follow the simple pattern that your examples follow. For example the multiplication tables of finite groups.2012-07-17

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In combinatorics, these are known as Latin Squares

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    Thanks! With this and Circulant Matrix, I'm not so alone with these beasts... Have to google and go study more...2012-07-17
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This reference, as well as this reference, uses the term block circulant matrix to refer to matrices with the structure you present.