  Motivated by a recent result of Elberfeld, Jakoby and Tantau showing that
$\mathsf{MSO}$ properties are Logspace computable on graphs of bounded
tree-width, we consider the complexity of computing the determinant of the
adjacency matrix of a bounded tree-width graph and as our main result prove
that it is in Logspace. It is important to notice that the determinant is
neither an $\mathsf{MSO}$-property nor counts the number of solutions of an
$\mathsf{MSO}$-predicate. This technique yields Logspace algorithms for
counting the number of spanning arborescences and directed Euler tours in
bounded tree-width digraphs.
  We demonstrate some linear algebraic applications of the determinant
algorithm by describing Logspace procedures for the characteristic polynomial,
the powers of a weighted bounded tree-width graph and feasibility of a system
of linear equations where the underlying bipartite graph has bounded
tree-width.
  Finally, we complement our upper bounds by proving $\mathsf{L}$-hardness of
the problems of computing the determinant, and of powering a bounded tree-width
matrix. We also show the $\mathsf{GapL}$-hardness of Iterated Matrix
Multiplication where each matrix has bounded tree-width.
