If M is diagonalizable, it is I, -I, or similar to diag(1,-1), and these satisfy M^2 = I.  If M is not diagonalizable, it has Jordan form [[1,1],[0,1]] or [[-1,1],[0,-1]], and these do not satisfy M^2=I.  So the the first case provides all solutions.   --  Gene From: David Wilson <davidwwilson@comcast.net> To: 'math-fun' <math-fun@mailman.xmission.com> Sent: Saturday, January 30, 2016 11:33 AM Subject: [math-fun] Matrix question Which 2x2 matrices M satisfy M^2 = I? Is there an approach to solving this that doesn't involve algebra on the matrix elements?