We use standard techniques of linear algebra to construct an infinite family of identities that involve finite weighted sums of the Möbius and Mertens functions, where the weights are equal to −1, 0, or 1. In a related manner, we construct, for each positive integer n, an n × n symmetric unimodular matrix, and each matrix is used to express an identity that involves a finite weighted sum of the Möbius function. We establish several results on the spectral decomposition of these matrices.