The Delassus matrix, closely related to the operational-space inertia matrix, is a fundamental quantity in robotics with applications in simulation, system identification, and control. Traditional approaches compute and store this matrix explicitly, either in sparse or dense form. In this work, we depart from this convention by treating the Delassus matrix as a matrix-free operator. We derive efficient algorithms with low computational complexity that multiply the Delassus matrix or its damped inverse by an input vector or matrix. Unlike approaches based on explicit matrix construction, our method achieves a linear memory footprint, making it scalable to problems with thousands of constraints and suitable for execution on resource-limited hardware. We implement these matrix-free operations on top of the open-source Pinocchio library and evaluate their performance against state-of-the-art methods that rely on explicit matrix computation. Our benchmarks demonstrate substantial speedups, ranging from approximately

$2\times$

$4\times$

for single-robot scenarios, up to

$10\times$

when joint friction is enabled, and over

$400\times$

in the Pyramid+Robots contact-rich multi-body scene.

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