The minimum rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose (i, j)th entry ( for i not equal j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. A universally optimal matrix is defined to be an integer matrix A such that every off-diagonal entry of A is 0, 1, or -1, and for all fields F, the rank of A is the minimum rank over F of its graph. Universally optimal matrices are used to establish field independence of minimum rank for numerous graphs. Examples are also provided verifying lack of field independence for other graphs.