A generalized Jacobi matrix is a matrix of infinite size indexed over $\mathbb{Z}$ whose non-zero entries are placed around the diagonal. The Lie algebra of such matrices over the complex number field $\mathbb{C}$ plays an important role in KP Hierarchy. In this talk, I explain how one can determine the homology of the Lie algebras of such matrices (and their orthogonal/symplectic variants) over an associative unital $k$-algebra, where $k$ is a field of characteristic $0$. If possible, I may also explain its possible application. This talk is based on a collaboration with A. Fialowski.