Let M be an invertible matrix, and let lambda be an eigenvalue of M. Show that 1/lambda is an eigenvalue of M-1. Solution is eigenvalue so we have Mx=x for some nonzero vector x By multiplying both sides with M -1 we get M -1 Mx=M -1 x x=M -1 x Now is nonzero because if =0, then Mx=0, so M -1 Mx=x=0. But x was nonzero so we can divide by 1/x=M -1 x which shows 1/ is an eigenvalue of M -1 .