We discuss Toeplitz and associated matrices which have simple explicit expressions for their inverses. We first review existing results and generalize these where possible, including matrices with hyperbolic and trigonometric elements. In Section 2 we discuss and generalize the Fiedler matrix. In Section 3 we give an analytic procedure for inverting any band Toeplitz matrix, in Section 4 we invert a tridiagonal Toeplitz matrix with modified corner elements.
