The determinant is just the thing that satisfies those properties. It's a label we apply to a function that follows certain laws. You might be curious where tho

The determinant is just the thing that satisfies those properties. It's a label we apply to a function that follows certain laws. You might be curious where those laws come from, or why we want those laws to be satisfied, but that's a different question. You're looking for a deeper meaning than exists. Mathematical objects are labels applied to entities that satisfy a certain set of properties.