Consider the class of rational functions that are the result of dividing one linear function by another:
One can easily compute that, for x≠cd
ddx(a+bxc+dx)=bc−ad(c+dx)2≶0 as ad−bc≷0
Thus, we can easily check whether such a rational function is increasing or decreasing (on any connected interval in its domain) by checking the determinant of a corresponding matrix
This made me wonder whether there is some known deeper principle that is behind this connection between linear algebra and rational functions (seemingly distant topics), or is this probably just a coincidence?
I’ll put it more simply. If the determinant is zero, then the linear functions ax+b and cx+d are rendered linearly dependent. For a pair of monovariant functions this forces the ratio between them to be a constant. The zero determinant condition is thereby a natural boundary between increasing and decreasing functions.