This paper is designed to pique the interest of undergraduate students who are familiar with the concepts of linear algebra. We investigate five methods of computing square roots of two-by-two matrices. Each method gives rise to applications and examples. Topics touched upon include solutions to Abel's functional equation, Fibonacci numbers, Mobius transformations, systems of differential equations, Newton's method applied to matrices (including surprising pictures and open questions), continued fraction representations of matrices, quadratic number fields, and quadratic forms.
Northshield, S. (2010). Square Roots of 2x2 Matrices. Contemporary Mathematics, 517, 289-304.