We study point-line configurations, their minimal matroids, and their associated circuit varieties. We present an algorithm for identifying the minimal matroids of these configurations with respect to dependency order, or equivalently, the maximal matroids with respect to weak order, and use it to determine the irreducible decomposition of their corresponding circuit varieties. Our algorithm is applied to several classical configurations, including the Fano matroid, affine plane of order three, MacLane, and Pappus configurations. Additionally, we explore the connection to a conjecture by Jackson and Tanigawa, which provides a criterion for the uniqueness of the minimal matroids.