Determinant and volume

Summary

The determinant $det (A)$ of a square matrix $A$ is the signed factor by which $A$ scales $n$ -dimensional volumes. Geometrically, $∣ det (A) ∣$ is the volume of the parallelepiped spanned by $A$ ‘s column vectors; the sign flips if $A$ contains a reflection.

Determinants tell you whether a linear map is invertible (non-zero det) or singular (zero det). They appear in change-of-variables formulas (probability density transformations, normalizing flows), in computing volumes of parallelotopes (Gaussian likelihoods), and as Jacobian determinants in differential geometry.

Properties

For $A, B \in R^{n \times n}$ :

$det (A B) = det (A) det (B)$ . Composition multiplies volumes.
$det (A^{⊤}) = det (A)$ .
$det (A^{- 1}) = 1/ det (A)$ when $A$ is invertible.
$det (c A) = c^{n} det (A)$ for scalar $c$ .
$det = 0 ⟺$ columns linearly dependent $⟺$ $A$ not invertible $⟺$ $A$ has a zero eigenvalue.
$det (A) = \prod_{i} λ_{i}$ (product of eigenvalues, with multiplicity, possibly complex).
For triangular matrices: $det = \prod_{i} A_{ii}$ (product of diagonal).

Geometric interpretation

The columns of $A$ are vectors in $R^{n}$ . They span a parallelepiped. Its volume is $∣ det (A) ∣$ .

$det (A) > 0$ : $A$ preserves orientation.
$det (A) < 0$ : $A$ reverses orientation (contains a reflection).
$det (A) = 0$ : parallelepiped is flat. Columns are linearly dependent. $A$ collapses at least one dimension.

For an orthogonal matrix $Q$ : $∣ det (Q) ∣ = 1$ (rotation/reflection preserves volume).

Learning objective

Separate area magnitude from orientation sign, then see why dependent columns have zero area.

Read it this way: compare the first two panels before looking at the third. Swapping the ordered columns leaves the parallelogram and its area unchanged but reverses orientation, so only the sign flips. When one column becomes a multiple of the other, the shape loses a dimension; its area and determinant both become zero.

Original coordinate construction checked against MIT OpenCourseWare 18.06, the Deep Learning linear algebra chapter, and the open Interactive Linear Algebra determinant text.

Change of variables (probability)

If $Y = f (X)$ with $f$ invertible and differentiable, the density of $Y$ is

p_{Y} (y) = p_{X} (f^{- 1} (y)) \cdot ∣ det J_{f^{- 1}} (y) ∣

where $J_{f^{- 1}}$ is the Jacobian of the inverse transform. This is the basis of normalizing flows: pick $f$ so the Jacobian determinant is cheap to compute (triangular Jacobian → $det = \prod_{i} \partial y_{i} / \partial x_{i}$ ).

Computing determinants

Method	Cost	When
LU decomposition	$O (n^{3})$	General-purpose; standard library default
Triangular: product of diag	$O (n)$	When matrix is already triangular
Eigendecomposition: $\prod λ_{i}$	$O (n^{3})$	If you need eigenvalues anyway
Log-determinant for PD matrices	via Cholesky	When you only need $lo g det$ (e.g., Gaussian log-likelihood)

Numerical tip: for a general matrix, use a signed log-determinant: track the row-pivot sign and the signs of the LU diagonal entries, then sum $lo g ∣ U_{ii} ∣$ . For a positive-definite matrix with $A = L L^{⊤}$ , compute $lo g det (A) = 2 \sum_{i} lo g L_{ii}$ . The determinant itself overflows or underflows quickly.

Common pitfalls

Using $det$ as a proxy for matrix “size.” A nearly singular matrix can have huge entries but tiny $det$ .
Computing $det$ for invertibility tests. Numerically unstable; use rank or condition number instead.
Forgetting the absolute value in change-of-variables formulas. Densities are non-negative; the Jacobian determinant can be negative.