I read in some lecture notes that "as an example for the intersection between linear algebra and calculus, one can establish the relationship between trace and determinant of a matrix using a density-argument".
Which relationship is meant? And what would the argument be?
For example, it can be seen that the set of diagonalizable matrices is dense in the set of $n \times n$ matrices over $\mathbb{C}$. So, when you have a continuous operation on $n \times n$ matrices, you can look at what it does to diagonal matrices and then argue that any other matrix is arbitrarily close to a diagonal matrix in some basis and then use limits to prove the result. For example, $\mathrm{det}$ is continuous on $n \times n$ matrices because the formula is given by a multivariate polynomial. Can you proceed from here?