Does exchanging rows and columns change the value of a functional determinant by a sign?

Question

Suppose we have an operator (with some suitable boundary conditions) like

\begin{equation} O =\begin{pmatrix} a(t) & \frac{d}{dt} +b(t) \\ \frac{d}{dt}+ b(t) & c(t) \end{pmatrix} \end{equation}

Would an exchange of rows or columns change the value of the functional determinant in the same way it would change the value of a regular determinant?