# Two by Two Matrices with Both Eigenvalues in $$Z/pZ$$

by Michael P. Knapp Loyola College)

Mathematics Magazine
April, 2006

Subject classification(s): Algebra and Number Theory | Algebra | Matrices
Applicable Course(s): 3.8 Linear/Matrix Algebra

This article provides a non-group theory approach to finding the number of two by two matrices over $$Z/pZ$$ that have both eigenvalues in the same field.  The strategy is to use the quadratic formula to find the roots of the characteristic polynomial of a matrix and then count the number of matrices for which these roots are in $$Z/pZ$$.

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