Determinants¶
Recall (doc/postdoc). Not a first course.
Job: unique alternating multilinear volume form on columns with \(\det I=1\); \(\mathrm{GL}_n=\{A:\det A\neq0\}\).
Core¶
\[
\det A=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_{i=1}^n a_{i,\sigma(i)}
\]
| Property | Formula |
|---|---|
| Multilinearity / alt. | \(\det\) alternating in columns |
| Multiplicativity | \(\det(AB)=\det A\det B\) |
| Transpose | \(\det A^T=\det A\) |
| Inverse | \(\det A\neq0\Leftrightarrow A\) invertible; \(\det(A^{-1})=(\det A)^{-1}\) |
| Eigenvalues | \(\det A=\prod\lambda_i\) (alg. closed, with mult.) |
| \(2\times2\) | \(ad-bc\) |
Cramer / expansion: Laplace expansion along row/column. Similar matrices: same det.
Orientation / volume: \(|\det A|=\) parallelepiped volume of columns; \(\operatorname{sign}\det\) orientation.
Used later for¶
- Matrix inversion: adjugate formula
- Eigenvalues: \(\chi_A(t)=\det(tI-A)\)
- Multivariable calculus: Jacobian
- Algebraic geometry / schemes: resultant, det bundles
Check¶
Is \(\det(A+B)=\det A+\det B\)?
No — only multilinear in each slot separately; counterexample \(I+I\) vs \(1+1\) in dim \(>1\).