Determinants

Definition

For $2\times 2$ and $3 \times 3$ matrices:

$$\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc,$$ $$\det\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg).$$

In general, cofactor (Laplace) expansion along row $i$:

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

where the minor $M_{ij}$ is the determinant with row $i$ and column $j$ deleted. Any row or column can be used — choose the one with the most zeros. (The full Leibniz formula sums over all $n!$ permutations, which is why cofactor expansion costs $O(n!)$ and elimination, at $O(n^3)$, is used in practice.)

Properties

Adjugate and inverse

The adjugate is the transpose of the cofactor matrix: $(\operatorname{adj}A)_{ij} = C_{ji}$. Then

$$A\,\operatorname{adj}A = \operatorname{adj}A\,A = (\det A)\,I \quad\Longrightarrow\quad A^{-1} = \frac{\operatorname{adj}A}{\det A}.$$

Consequences for $n\times n$ $A$: $\det(\operatorname{adj}A) = (\det A)^{n-1}$; $\operatorname{adj}(\operatorname{adj}A) = (\det A)^{n-2}A$; $\operatorname{adj}(AB) = \operatorname{adj}B\,\operatorname{adj}A$.

Geometric meaning

$\lvert\det A\rvert$ is the factor by which $A$ scales areas ($n = 2$) or volumes ($n = 3$): the unit square maps to a parallelogram of area $\lvert\det A\rvert$. The sign says whether orientation is preserved. $\det A = 0$ means the space is flattened into a lower dimension — information is lost and $A$ is not invertible.

• Area of the parallelogram spanned by $\mathbf u, \mathbf v \in \mathbb R^2$: $\lvert\det[\mathbf u\ \mathbf v]\rvert$; triangle: half of that.
• Volume of the parallelepiped spanned by $\mathbf u, \mathbf v, \mathbf w$: $\lvert\det[\mathbf u\ \mathbf v\ \mathbf w]\rvert$ (the scalar triple product).
• In multivariate calculus and probability, the Jacobian determinant rescales densities under change of variables.

Special determinants

Vandermonde:

$$\det\begin{pmatrix}1 & x_1 & x_1^2 & \cdots & x_1^{n-1}\\ \vdots & & & & \vdots\\ 1 & x_n & x_n^2 & \cdots & x_n^{n-1}\end{pmatrix} = \prod_{1 \le i < j \le n}(x_j - x_i).$$

Non-zero iff all $x_i$ are distinct — which is why polynomial interpolation through distinct points has a unique solution.

Block triangular: $\det\begin{pmatrix}A&B\\0&D\end{pmatrix} = \det A\det D$.

Rank-one update (matrix determinant lemma): $\det(I + \mathbf u\mathbf v^\top) = 1 + \mathbf v^\top\mathbf u$; e.g. the $n \times n$ matrix with $a$ on the diagonal and $b$ elsewhere has determinant $(a - b)^{n-1}(a + (n-1)b)$.

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