Does a nested matrix have any mathematical meaning?

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Say A and B are two 2x2 matrices, and C is defined below. $$ C= \begin{bmatrix} A & B\\ B & A \end{bmatrix} $$ Is this nonsensical or does it have a meaning? If we're talking about nested determinants, it makes sense. I was wondering if nested matrices are gibberish or have any mathematical meaning.