In this paper we demonstrate the capabilities of geometric algebra by the derivation of a formula for the determinant of the sum of two matrices in which both matrices are separated in the sense that the resulting expression consists of a sum of traces of products of their compound matrices. For the derivation we introduce a vector of Grassmann elements associated with an arbitrary square matrix, we recall the concept of compound matrices and summarize some of their properties. This paper introduces a new derivation and interpretation of the relationship between p-forms and the pth compound matrix, and it demonstrates the use of geometric algebra, which has the potential to be applied to a wide range of problems.