A method of fabricating a ceramic matrix composite includes generating a stream of vaporized precursor and, optionally, a vaporized rare earth element. The vaporized precursor is a precursor of either silicon carbide or silicon nitride. The stream flows for one or more periods of time through a chamber that contains a fibrous structure such that the fibrous structure is exposed to the stream. The fibrous structure initially contains no silicon carbide matrix or silicon nitride matrix. The vaporized precursor deposits over the period of time on the fibrous structure as a substantially fully dense ceramic matrix of either the silicon carbide or the silicon nitride. For at least a portion of the period of time, the vaporized rare earth element is included in the stream such that the ceramic matrix deposited during that time includes dispersed rare earth element.