  In this paper, we address the weighted linear matroid intersection problem
from the computation of the degree of the determinants of a symbolic matrix. We
show that a generic algorithm computing the degree of noncommutative
determinants, proposed by the second author, becomes an $O(mn^3 \log n)$ time
algorithm for the weighted linear matroid intersection problem, where two
matroids are given by column vectors $n \times m$ matrices $A,B$. We reveal
that our algorithm is viewed as a "nonstandard" implementation of Frank's
weight splitting algorithm for linear matroids. This gives a linear algebraic
reasoning to Frank's algorithm. Although our algorithm is slower than existing
algorithms in the worst case estimate, it has a notable feature: Contrary to
existing algorithms, our algorithm works on different matroids represented by
another "sparse" matrices $A^0,B^0$, which skips unnecessary Gaussian
eliminations for constructing residual graphs.
