<p>This paper presents a novel design framework for photonic matrix multiplication based on programmable photonic integrated circuits that use double racetrack (DRT) resonators as building blocks. Here, we analytically demonstrate that the transfer function of the DRT resonator building block resembles that of conventional building blocks, such as directional couplers and MZI, making it suitable for implementing programmable circuits that handle complex matrix calculations. Using this new DRT resonator building block, a 3-by-3 photonic processor is implemented and validated through full-wave Finite Element Method (FEM) simulations, and scalability is further analysed using hybrid FEM-circuit modelling. Additionally, we implement a low-pass filter as a non-unitary system example, showcasing the flexibility of the approach. Results confirm high fidelity between simulated and analytical models, supporting the viability of DRT resonators for reconfigurable photonic circuits. We believe that the proposed DRT resonator building blocks have the potential to complement and integrate with other previously reported blocks, thereby enhancing fidelity and expanding the application scope of programmable photonic integrated circuits, particularly for all-optical signal processing in communication systems and for integration within microwave photonics platforms targeting emerging telecommunications technologies.</p>