If you'd like to know where your sweatheart has been, no need to setup any spy GPS tracker on her/his car... All you have to do is to take a peek at her/his phone.
Let's take iPhone for example. Do you guys know what "Significant Locations" is? You will be surprised to learn that iOS stores the time and place of locations you've visited.
Take a look at items in "HISTORY" and you will see that it is a minute by minute tracking tool that stores and logs all of your activity.
Fortunately this feature is enabled by default and that the setting is hidden away such that the average user probably doesn't find it. If Suspect's iPhone is unlocked, don't forget to take a look at her/his location history. Maybe you could get more clue which can give you some insight into where suspect really was when they said they were sleeping at home.