A Hyperbolic Metric

The Poincare half-plane model for hyperbolic space puts the following metric on the plane:

\[ds^2 = \frac{1}{y^2} (dx^2 + dy^2).\]

Compute the Ricci scalar \(R\) for this metric in matrix form. Is this a vacuum solution to Einstein's equations? Give your answer as an (R, Yes/No) pair.

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