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Two by Two Matrices with Both Eigenvalues in \(Z/pZ\)

by Michael P. Knapp Loyola College)

This article originally appeared in:
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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Capsule Course Topic(s):
Linear Algebra | Characteristic Polynomial
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