Let z and w be distinct points of the bounded domain of holomorphy U in \({\mathbb{C}}^ d\). For each d-tuple M of commuting \(2\times 2\) complex matrices with joint spectrum \(\{\) z,w\(\}\), let \(\theta_ M\) denote the angle between its two (one-dimensional) eigenspaces. The author proves that the Carathéodory distance is given by \[ c_ U(z,w)=\inf \tanh^{-1}(\sin \theta_ M), \] where the infimum is taken over all such M satisfying \(\| h(M)\| \leq \| h\|_{\infty}\) whenever h is a bounded holomorphic function on U. Here \(\| h\|_{\infty}\) denotes the supremum norm, and \(\| h(M)\|\) denotes the operator norm of the \(2\times 2\) matrix h(M) regarded as a linear transformation of \({\mathbb{C}}^ 2.\) The author also provides an analogous formula for the Carathéodory- Reiffen metric. These formulas allow the author to apply complex geometry to operator theory and vice versa. Thus he uses the Carathéodory distance on \(n\times n\) matrix balls to prove that certain contractive unital representations of function algebras are completely contractive, generalizing a result of \textit{V. Paulsen} [J. Oper. Theory 18, No.2, 249- 263 (1987; Zbl 0655.47008)]. Conversely, this result and the dilation techniques of operator theory lead to a new proof for \textit{L. Lempert}'s theorem [Anal. Math. 8, 257-261 (1982; Zbl 0509.32015)] about equality of the Carathéodory and Kobayashi distances when U is convex. Finally the author indicates how Lempert's theorem can be regarded as the case \(n=2\) for interpolating n distinct points of a bounded convex domain by a totally geodesic analytic disc.