# 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?