Department of Mathematics
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The eight "phases" of a general square matrix resulting from rotations and reflections may be separated into two sets which interlace each other. The effects of rotation (e.g., a quarter turn) have the effect of moving between the two sets. We have found two theorems which describe:

1. the sign of the determinant (if non-singular), and
2. a pair of characteristic equations (and thus eigenvalues).

Simple second order examples suffice to illustrate the theorems, which emerged from a study of higher order matrices of magic squares. Also the Cayley-Hamilton theorem may be useful in this context, and we note that it holds for square matrices over commutative rings as well.