Determinant

81. Determinant

A scalar attached to every square matrix that measures the signed scale factor the matrix applies to areas / volumes when used as a linear map.

Notation: $det (𝐴)$ , or $| 𝐴 |$ .

81.1. From first principles: the $2 \times 2$ case

Take two vectors $\vec{𝑎}, \vec{𝑏} \in ℝ^{2}$ :

\vec{𝑎} = [\begin{matrix} 𝑎_{1} \\ 𝑎_{2} \end{matrix}] \vec{𝑏} = [\begin{matrix} 𝑏_{1} \\ 𝑏_{2} \end{matrix}]

They span a parallelogram. Its signed area is:

Area = 𝑎_{1} 𝑏_{2} - 𝑎_{2} 𝑏_{1}

That signed area is exactly the determinant of the matrix whose columns are $\vec{𝑎}, \vec{𝑏}$ :

det [\begin{matrix} 𝑎_{1} & 𝑏_{1} \\ 𝑎_{2} & 𝑏_{2} \end{matrix}] = 𝑎_{1} 𝑏_{2} - 𝑎_{2} 𝑏_{1}

Sign:

$+$ if $\vec{𝑏}$ is counterclockwise from $\vec{𝑎}$ (preserved orientation)
$-$ if clockwise (orientation flipped)
$0$ if $\vec{𝑎}$ and $\vec{𝑏}$ are collinear (parallelogram degenerates)

Example

𝐴 = [\begin{matrix} 3 & 1 \\ 0 & 2 \end{matrix}] det (𝐴) = 3 \cdot 2 - 1 \cdot 0 = 6

The unit square ${[0, 1]}^{2}$ maps to a parallelogram of area $6$ — six times bigger.

𝐵 = [\begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix}] det (𝐵) = 1 \cdot 4 - 2 \cdot 2 = 0

The columns are parallel (one is twice the other) — image collapses to a line, area $0$ .

81.2. $3 \times 3$ : signed volume of the parallelepiped

Three column vectors $\vec{𝑎}, \vec{𝑏}, \vec{𝑐} \in ℝ^{3}$ span a parallelepiped. Its signed volume is:

det [\begin{matrix} 𝑎_{1} & 𝑏_{1} & 𝑐_{1} \\ 𝑎_{2} & 𝑏_{2} & 𝑐_{2} \\ 𝑎_{3} & 𝑏_{3} & 𝑐_{3} \end{matrix}] = 𝑎_{1} (𝑏_{2} 𝑐_{3} - 𝑏_{3} 𝑐_{2}) - 𝑎_{2} (𝑏_{1} 𝑐_{3} - 𝑏_{3} 𝑐_{1}) + 𝑎_{3} (𝑏_{1} 𝑐_{2} - 𝑏_{2} 𝑐_{1})

This is the cofactor expansion along the first column — three $2 \times 2$ determinants, alternating sign.

81.3. General $𝑛 \times 𝑛$ : cofactor expansion

For an $𝑛 \times 𝑛$ matrix $𝐴$ , expand along any row $𝑖$ (or any column $𝑗$ ):

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

where $𝑀_{𝑖 𝑗}$ is the minor — the determinant of the $(𝑛 - 1) \times (𝑛 - 1)$ submatrix obtained by deleting row $𝑖$ and column $𝑗$ of $𝐴$ (see Minor).

The factor ${(- 1)}^{𝑖 + 𝑗}$ produces the checkerboard sign pattern:

[\begin{matrix} + & - & + & \dots \\ - & + & - & \dots \\ + & - & + & \dots \\ ⋮ & ⋮ & ⋮ & ⋱ \end{matrix}]

The recursion bottoms out at the $1 \times 1$ case: $det ([𝑎]) = 𝑎$ .

81.4. Key properties

Identity: $det (𝐼_{𝑛}) = 1$
Multiplicativity: $det (𝐴 𝐵) = det (𝐴) det (𝐵)$
Transpose: $det (𝐴^{𝑇}) = det (𝐴)$
Inverse: $det (𝐴^{- 1}) = \frac{1}{det (𝐴)}$ (when defined)
Scalar multiple: $det (𝑐 𝐴) = 𝑐^{𝑛} det (𝐴)$ for an $𝑛 \times 𝑛$ matrix
Triangular matrix: $det =$ product of diagonal entries
Row swap: flips the sign
Row scaled by $𝑐$ : multiplies $det$ by $𝑐$
Row $+$ multiple of another row: leaves $det$ unchanged (basis of Gaussian-elimination computation)

81.5. What $det (𝐴)$ tells you about $𝐴$

$det (𝐴)$	Implication
$\neq 0$	$𝐴$ is invertible (see Matrix Inverse); columns are linearly independent (see Linear Independence); columns span $ℝ^{𝑛}$ ; $𝐴$ has full rank (see Rank); $𝐴 𝑥 = 𝑏$ has a unique solution for every $𝑏$ .
$= 0$	$𝐴$ is singular — non-invertible; columns are linearly dependent; image is a strict subspace; $𝐴 𝑥 = 0$ has non-trivial solutions; the linear map collapses dimension.

81.6. Geometric interpretation

For a linear map $𝑇 : ℝ^{𝑛} \to ℝ^{𝑛}$ given by matrix $𝐴$ :

$| det (𝐴) |$ = factor by which $𝑇$ scales $𝑛$ -dimensional volume
$sign (det (𝐴))$ = whether $𝑇$ preserves $(+)$ or reverses $(-)$ orientation

Example

Reflection across the $𝑦$ -axis:

𝑅 = [\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}] det (𝑅) = - 1

Areas preserved (factor $| - 1 | = 1$ ); orientation reversed (sign).

81.7. Computing $det$ in practice

$2 \times 2$ / $3 \times 3$ : direct formula above.
Larger matrices: cofactor expansion is $𝑂 (𝑛!)$ — impractical past $𝑛 \approx 10$ .
Standard method: row-reduce to triangular form, multiply diagonal entries, track row-operation effects on $det$ (see Gaussian Elimination). This is $𝑂 (𝑛^{3})$ .
LU decomposition (see LU Decomposition): $det (𝐴) = det (𝐿) det (𝑈) = \prod$ of $𝑈$ ’s diagonal entries (since $𝐿$ has unit diagonal).

81.8. Connection to other concepts

Minor — building block for cofactor expansion
Adjugate — matrix of cofactors, used in $𝐴^{- 1} = \frac{adj (𝐴)}{det (𝐴)}$
Eigenvectors & Eigenvalues — eigenvalues are roots of $det (𝐴 - 𝜆 𝐼) = 0$
Unimodularity — matrices with $det = \pm 1$

81. Determinant

81.1. From first principles: the 2×2 case

81.2. 3×3: signed volume of the parallelepiped

81.3. General 𝑛×𝑛: cofactor expansion