The Square Roots of 2 × 2 Matrices
D. Sullivan · Mathematics Magazine · 1993
Introduction In a recent article MacKinnon [1] describes four methods that may be used to find square roots of 2 X 2 matrices. The first of these methods requires that the matrix for which the square roots are sought be diagonalizable and, subsequently, this method was used by Scott [2] to determine all the square roots of 2 X 2 matrices. A surprising conclusion is that scalar 2 X 2 matrices possess double-infinities of square roots whereas nonscalar 2 X 2 matrices have only a finite number of square roots. The purpose of this article is to show how the Cayley-Hamilton theorem may be used to determine explicit formulae for all the square roots of 2 X 2 matrices. These formulae indicate exactly when a 2 X 2 matrix has square roots, and the number of such roots. By definition, the square roots of a 2 X 2 matrix, A, are those 2 X 2 matrices, X, for which