Reflecting the Pascal matrix about its main antidiagonal

Let B denote either of two varieties of order n Pascal matrix, i.e., one wh ose entries are the binomial coefficients. Let B-R denote the reflection of B about its main antidiagonal. The matrix B is always invertible module n; our main result asserts that B-1 = B-R mod n if and only if n is prime. In the course of motivating this result we encounter and highlight some of th e difficulties with the matrix exponential under modular arithmetic. We the n use our main result to extend the "Fibonacci diagonal" property of Pascal matrices.