1. The problem statement, all variables and given/known data Prove that the only invertible nxn idempotent matrix is the identity matrix. 2. Relevant equations idempotent: given an nxn matrix, A^2=A invertible: There exists a matrix B such that AB=BA=I I, or identity matrix: for all matrices A, AI=IA=A 3. The attempt at a solution So far I haven't gotten very far: Suppose A is an nxn matrix. For A to be invertible there exists a matrix B such that AB=BA=I. For A to be idempotent A^2=A. Otherwise, I've kind of mulled it over and gotten this far: To be invertible it has to have a determinant != 0. So, I think that I have to show that the determinant of everything else that is indempotent is 0, which would mean widely defining the criteria for a matrix to be indempotent and showing that the two are mutually exclusive. The problem is I'm not quite sure about the properties of indempotent matrices and, since we don't have a formula for the determinant of a given nxn matrix, I'm not quite sure what to do. So far, I have not covered and am not allowed to use rank or eigenvalues in my class. Thanks!