Lesson 34 — 2x2 and 3x3 Determinants

Computation and properties

2x2

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

3x3 — Sarrus's rule

$\det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = aei + bfg + cdh - ceg - bdi - afh$

(Repeat the first 2 columns to the right, 3 descending products − 3 ascending products.)

n×n — Laplace expansion (cofactors)

$\det A = \sum_{j=1}^n (-1)^{i+j} a_{ij} M_{ij}$

where $M_{ij}$ is the minor (det of the submatrix removing row $i$ and column $j$). Recursive: reduces $n \times n$ to a sum of $(n-1) \times (n-1)$.

Definition via permutations (Leibniz)

$\det A = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}$

Sum over all $n!$ permutations.

Properties

Geometric interpretation

• $|\det A|$ = volume of the parallelepiped generated by the columns of $A$.
• $\det A > 0$: orientation preserved. $\det A < 0$: orientation reversed.
• $\det A = 0$: linearly dependent columns ("flattened" parallelepiped).

Invertibility criterion

$A$ invertible $\iff \det A \neq 0$.