Every minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field FF\mathbb F, the list contains only finitely many FF\mathbb F-representable matroids, due to the well-quasi-ordering of FF\mathbb F-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these FF\mathbb F-representable excluded minors in general. We consider the class of matroids of path-width at most kkk for fixed kkk. We prove that for a finite field FF\mathbb F, every FF\mathbb F-representable excluded minor for the class of matroids of path-width at most~kkk has at most 2|F|O(k2)2|F|O(k2)2^{|\mathbb{F}|^{O(k^2)}} elements. We can therefore compute, for any integer kkk and a fixed finite field FF\mathbb F, the set of FF\mathbb F-representable excluded minors for the class of matroids of path-width kkk, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an FF\mathbb F-represented matroid is at most kkk. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most kkk has at most 22O(k2)22O(k2)2^{2^{O(k^2)}} vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.