Determinants

Determinants & Formulas

The determinant is a scalar value encoding a square matrix's invertibility, volume scaling, and orientation.

For a $2 \times 2$ matrix: $$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$$

For $3 \times 3$, use cofactor expansion along the first row: $$\det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$$ where $C_{ij} = (-1)^{i+j}M_{ij}$ and $M_{ij}$ is the minor.

Key properties:

Geometry & Pitfalls

Geometrically, absolute determinants measure space scaling:

A negative determinant means the transformation reverses orientation.

Physics link: The Jacobian determinant $\det(\partial x_i/\partial q_j)$ handles coordinate changes in integrals.

Common pitfall: $\det(A+B) \neq \det A + \det B$. Determinants respect multiplication, but never addition.