Before teaching the chapter on determinants in a linear-algebra course for beginning undergraduate students (mathematics and computer science, more specifically) I would like to give a small introduction and convince the students that this is an interesting notion.
What could one tell beginning undergraduate students in 15 to 30 minutes to get them more interested in determinants?
What then would follow are the "usual things" (special cases for small dimension, multi-linearity, computation via row and column transform, computation by developing along a line/column, Leibniz formula, Cramer's rule and formula for the inverse matrix).
I might modify this slightly if it helps to make the motivation make sense in retrospect, but to the extent possible I would like to keep the question focused on an introductory presentation (rather than the actual content), as I am aware of the question "Is there a good way to explain determinants in an elementary linear algebra class?" which covers the aspect how to teach determinants.
To high-light and to reiterate, the intended difference between the two questions is that while the referenced one asks about how to actually teach the subject I am mainly interested in advice for a short (non-technical) introduction to be given before the subject of determinants is actually discussed. Thus, I would be willing to do some hand-waving and over-simplifications to convey an application of determinants and the related ideas also in areas currently beyond the students reach.
Something I would like to say is: Knowing the determinant is important to doing ThisInterestingThing?
I am mainly looking forward to answers in this sense, but more general considerations are welcome, too.