  Compound matrices play an important role in many fields of mathematics and
have recently found new applications in systems and control theory. However,
the explicit formulas for these compounds are non-trivial and not always easy
to use. Here, we derive new formulas for the multiplicative and additive
compounds of a matrix using Kronecker products and sums. This provides a new
approach to matrix compounds based on the well-known and powerful theory of
Kronecker products and sums. We demonstrate several applications of these new
formulas, including deriving a new expression for the additive compound of the
product of two matrices.
