Suggestions for Determinant from Matrix Multiplication

Ideas for further contemplation from the blog post Determinant from Matrix Multiplication.

1. For $n \times n$ matrices, use the Laplace Expansion (which was probably how you learned to calculate determinants in school, anyway).
2. Derive a general formula for $n \times n$ matrices using Mathematical Induction.
3. Determinants for non-square matrices?
4. Are there consequences for the eigenvalues?