[en] A family of matrix theories of elastic wave scattering is derived, and one, which is in a certain sense optimal, is developed. Called the method of optimal truncation (MOOT), it results from a minimum principle and can be shown to yield a convergent sequence of approximations. Numerical results for scattering cross-sections for longitudinal incident waves with ka less than or equal to 10 from fixed rigid obstacles and voids with axial symmetry are obtained using MOOT, and are compared with results of other matrix theories. Shapes considered include spheres, oblate and prolate spheroids, pillboxes, and cones. Convergence is demonstrated. Extension of the method to elastic and fluid inclusions is discussed, as is its application to cracks, which may be accomplished by simulating the crack with an incompletely bonded identical inclusion. Implications of reciprocity and time-reversal invariance are discussed