It is the purpose of this paper to describe the homology of \(K_A\) (Koszul complex), in the case \(n=2\). The article is organized as follows. Section 2 contains some preliminaries together with the definition of the complexes \(B(a,b;A)\). Section 3 shows that over \(\mathbb{C}\), the bihomogeneous components of \(K_A\) are expressible in terms of the complexes \(B(a,b;A)\). Section 4 completely describes all \(H_\bullet(B(a,b;A))\), under the assumption that \(A\) is a square and invertible matrix. Section 5 derives from Section 4 a description of \(H_\bullet (K_A)\) over \(\mathbb{C}\), in case \(A\) is invertible, and provides some clues on \(H_\bullet(K_A)\), when \(A\) is not a square invertible matrix.