Determinants

The determinant is a scalar value that encodes key information about a square matrix: invertibility, volume scaling, and orientation.

Formulas

For a $2\times 2$ matrix:

$$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$$

For a $3\times 3$ matrix, expand along the first row (cofactor expansion):

$$\det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$$

where $C_{ij} = (-1)^{i+j}M_{ij}$ and $M_{ij}$ is the minor.

Key properties

• $\det(AB) = \det(A)\cdot\det(B)$
• $\det(A^T) = \det(A)$
• $\det(\alpha A) = \alpha^n \det(A)$ for an $n\times n$ matrix
• A matrix is invertible $\iff$ $\det(A) \neq 0$

Geometric meaning

• In $\mathbb{R}^{2}$: $|\det(A)|$ = area of the parallelogram spanned by the column vectors.
• In $\mathbb{R}^{3}$: $|\det(A)|$ = volume of the parallelepiped.
• A negative determinant means the transformation reverses orientation.

Physics link: The Jacobian determinant $\det(\partial x_i/\partial q_j)$ appears whenever you change coordinates in integrals — from Cartesian to polar, spherical, or any generalised system.

Common pitfall: $\det(A+B) \ne \det A + \det B$. The determinant respects multiplication ($\det AB = \det A \det B$) but not addition — it is an area scale factor, and areas do not add across different maps.