Determinant and volume

One-line definition

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.

Why it matters

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

• $\det (AB)=\det (A)\det (B)$. Composition multiplies volumes.
• $\det (A^{\top })=\det (A)$.
• $\det (A^{-1})=1/\det (A)$ when $A$ is invertible.
• $\det (cA)=c^n\det (A)$ for scalar $c$.
• $\det =0⟺$ columns linearly dependent $⟺$ $A$ not invertible $⟺$ $A$ has a zero eigenvalue.
• $\det (A)={\prod }_i{\lambda }_i$ (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 ${\mathbb{R}}^n$. 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).

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: $\prod {\lambda }_i$	$O(n^3)$	If you need eigenvalues anyway
Log-determinant for PD matrices	via Cholesky	When you only need $\log \det$ (e.g., Gaussian log-likelihood)

Numerical tip: for large matrices, compute $\log \det$ directly (sum of $\log$ of LU diagonal); $\det$ 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.