*1. Derivative of a determinant*

Consider the determinant *W _{(t)} * of a

*n×n*matrix

*Y*which each element is a function of

*t*. Assume elements to be independent variables. Then we could write:

Where *C _{ij}* are the corresponding cofactors. Thus we have:

Let’s define γ* _{i}* as the new matrices form by substituting the ith row with its derivative. Then we could write (2) in a tidier form:

*2. Abel-Jacobi-Liouville identity *

As we know, any system of linear ordinary equations can be extracted in to a single linear equation, namely:

And it is followed by that,

So we obseverve that a particular **row** of the derivative is the linear combination of the original rows, since for the kth row of *Y*, different elements on jth column are multiplied by the same factor *A*_{ik}. So of course, each term on the right hand side of (3) will be *W* times the corresponding element of *A _{ii}*.

Thus,

This has resulted in some interesting conclusions. For example if the solutions are independent for any point within the domin, they must be independent entirely.

[Literature: Pontryagain 1962, Chapter 3]