Determinant from Matrix Multiplication

Another old idea:

2011-02-03

Given a 2 \times 2 matrix \mathbf{A},

\det(\mathbf{A}) = \begin{bmatrix}1&0\end{bmatrix}\mathbf{A}^\intercal\begin{bmatrix}0&1\\-1&0\end{bmatrix}\mathbf{A}\begin{bmatrix}0\\1\end{bmatrix}.

Prove this and extend it to general n \times n matrices. Can you go further than that?

Suggestions

  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?
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