Determinants

What the Determinant Measures

The determinant of a square matrix $A$, written $\det(A)$ or $|A|$, answers a geometric question: by what factor does the transformation $\vec{x} \mapsto A\vec{x}$ scale volumes?

More precisely: if $S$ is any region in $\mathbb{R}^n$ with volume $\text{vol}(S)$, then the image $A(S)$ has volume $|\det(A)| \cdot \text{vol}(S)$. The sign of $\det(A)$ captures orientation: positive means orientation-preserving, negative means orientation-reversing (like a reflection).

When $\det(A) = 0$, the transformation squashes space into a lower dimension — the output has zero volume, and $A$ is singular.

Properties

The determinant is characterized by three properties:

1. Multilinearity: $\det$ is linear in each row separately (with other rows held fixed)
2. Alternating: swapping two rows negates $\det$ — equivalently, $\det = 0$ if two rows are identical
3. Normalization: $\det(I) = 1$

These three properties uniquely determine the determinant. All other properties follow:

• $\det(A') = \det(A)$ — rows and columns interchangeable
• $\det(AB) = \det(A)\det(B)$ — determinant of a product is the product of determinants
• $\det(A^{-1}) = 1/\det(A)$ — follows from the product rule
• $\det(cA) = c^n \det(A)$ for $n \times n$ matrix
• Adding a multiple of one row to another leaves $\det$ unchanged — scaling by the pivot in Gauss's method multiplies $\det$ by that scalar

Computing via Row Reduction

Gauss's method gives the fastest route to the determinant for large matrices. Tracking the effect of each row operation:

Reduce $A$ to echelon form $U$. The determinant of $U$ is the product of its diagonal entries (the pivots). Then:

$$\det(A) = (-1)^s \cdot \frac{1}{k_1 k_2 \cdots} \cdot u_{11} u_{22} \cdots u_{nn}$$

where $s$ = number of row swaps and $k_i$ = scaling factors applied during reduction.

In practice: reduce to echelon form, track signs from swaps, multiply pivots.

The Permutation Expansion

For an $n \times n$ matrix, the determinant can be written as a sum over all $n!$ permutations $\sigma$ of $\{1,\ldots,n\}$:

$$\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma)\, a_{1,\sigma(1)}\, a_{2,\sigma(2)} \cdots a_{n,\sigma(n)}$$

where $\text{sgn}(\sigma) = +1$ if $\sigma$ is an even permutation (achievable by an even number of transpositions) and $-1$ if odd.

For $2 \times 2$: two permutations, $(1,2)$ with sign $+1$ and $(2,1)$ with sign $-1$, giving $\det = a_{11}a_{22} - a_{12}a_{21}$.

For $3 \times 3$: six permutations (Sarrus' rule). For larger $n$, the permutation expansion is impractical ($n!$ grows fast) but theoretically important — it proves properties of the determinant.

Laplace's Expansion (Cofactor Expansion)

For computational purposes, Laplace expansion along row $i$:

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

where $M_{ij}$ is the minor — the determinant of the $(n-1) \times (n-1)$ matrix obtained by deleting row $i$ and column $j$. The factor $(-1)^{i+j} M_{ij}$ is the cofactor $C_{ij}$.

Choosing a row or column with many zeros minimizes computation. Expansion along any row or column gives the same result.

Cramer's Rule

For an invertible $n \times n$ system $A\vec{x} = \vec{b}$, each component of the solution is a ratio of determinants:

$$x_i = \frac{\det(A_i)}{\det(A)}$$

where $A_i$ is $A$ with column $i$ replaced by $\vec{b}$.

Cramer's rule is elegant but computationally expensive ($O(n!)$ naively, or $O(n^4)$ with efficient determinant computation). For $n > 3$, Gaussian elimination is always faster. Cramer's rule is mainly useful for theoretical arguments and for $2 \times 2$ and $3 \times 3$ closed-form solutions.