**Statement of Liouville's Formula**

Consider the *n*-dimensional first-order homogeneous linear differential equation

on an interval *I* of the real line, where *A*(*x*) for *x* ∈ *I* denotes a square matrix of dimension *n* with real or complex entries. Let Φ denote a matrix-valued solution on *I*, meaning that each Φ(*x*) is a square matrix of dimension *n* with real or complex entries and the derivative satisfies

Let

denote the trace of *A*(*ξ*) = (*a*_{i, j }(*ξ*))_{i, j ∈ {1,...,n}}, the sum of its diagonal entries. If the trace of *A* is a continuous function, then the determinant of Φ satisfies

for all *x* and *x*_{0} in *I*.

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