Previous studies have found that participants use two cognitive strategies—constructive matching and response elimination—in responding to items in the Raven's Advanced Progressive Matrices (APM). This study proposed a multi-strategy psychometric model that builds on item responses and also incorporates eye-tracking measures, including but not limited to the proportional time on matrix area (PTM), the rate of toggling (ROT), and the rate of latency to first toggle (RLT). By jointly analyzing item responses and eye-tracking measures, this model can measure each participant's intelligence and identify the cognitive strategy used by each participant for each item in the APM. Several main findings were revealed from an eye-tracking-based APM study using the proposed model: (1) The effects of PTM and RLT on the constructive matching strategy selection probability were positive and higher for the former than the latter, while the effect of ROT was negligible. (2) The average intelligence of participants who used the constructive matching strategy was higher than that of participants who used the response elimination strategy, and participants with higher intelligence were more likely to use the constructive matching strategy. (3) High-intelligence participants increased their use of the constructive matching strategy as item difficulty increased, whereas low-intelligence participants decreased their use as item difficulty increased. (4) Participants took significantly less time using the constructive matching strategy than the response elimination strategy. Overall, the proposed model follows the theory-driven modeling logic and provides a new way of studying cognitive strategy in the APM by presenting quantitative results.