Determinants — Properties and Computation

The determinant is a scalar $det (A)$ assigned to every square matrix $A$ , encoding whether $A$ is invertible (nonzero $⟺$ invertible), the signed volume scaling of the linear map $A$ , and the orientation it induces. Key results: multiplicativity $det (A B) = det (A) det (B)$ , row-operation rules, cofactor (Laplace) expansion, and efficient computation via Gaussian elimination ( $O (n^{3})$ ). Used throughout linear algebra, geometry, differential equations, and physics.

The determinant is a function $det : M_{n} (F) \to F$ defined on square matrices over a field $F$ , uniquely characterized by being multilinear and alternating in the rows (or columns) and normalized so that $det (I) = 1$ . It matters because a single number simultaneously tells you whether a system $A x = b$ has a unique solution, how the associated linear transformation rescales volume, and the orientation behaviour of the map. A surprising fact: although the naive permutation-sum formula contains $n!$ terms — astronomically many even for moderate $n$ — the determinant can be computed in only $O (n^{3})$ arithmetic operations by row reduction, because the determinant's structural properties (not its closed form) drive the computation. This separation between definition and computation is the central theme of the topic.

The Core Insight

A determinant is the unique way to measure signed $n$ -dimensional volume of the parallelepiped spanned by the rows of a matrix, subject to three demands: it scales linearly in each row, it vanishes when two rows coincide (a degenerate, zero-volume box), and the unit cube has volume $1$ . Every property — multiplicativity, the effect of row operations, cofactor expansion — is a logical consequence of those three axioms. Computation reduces to triangularizing the matrix, where the determinant is just the product of the diagonal.

Formal Statement

Determinant via the Leibniz formula

For $A = (a_{ij}) \in M_{n} (F)$ , the determinant is

det (A) = σ \in S_{n} \sum sgn (σ) i = 1 \prod n a_{i, σ (i)},

where

S_{n}

is the symmetric group of all permutations of

{1, \dots, n}

, and

sgn (σ) \in {+ 1, - 1}

is the signature of

σ

(equal to

(- 1)^{k}

where

k

is the number of transpositions composing

σ

⟳This is the only multilinear alternating form normalized at

I

; each term picks exactly one entry from each row and column, weighted by the parity of the permutation matching rows to columns.

Determinant via axioms (Weierstrass / Artin)

The determinant is the unique function $D : M_{n} (F) \to F$ satisfying, with $A$ written by its rows $r_{1}, \dots, r_{n}$ :

(i) Multilinearity: linear in each row separately,

D (\dots, α r_{i} + β r_{i}^{'}, \dots) = α D (\dots, r_{i}, \dots) + β D (\dots, r_{i}^{'}, \dots) .

⟳fixing all other rows, the map

r_{i} \mapsto D

is a linear functional, reflecting the additivity of signed volume along one edge.

(ii) Alternating: $D = 0$ whenever two rows are equal.

(iii) Normalization: $D (I) = 1$ .

These three force $D = det$ , the function in equation [1].

Fundamental properties

For $A, B \in M_{n} (F)$ :

det (A^{T}) = det (A) .

⟳in [1], reindex by

τ = σ^{- 1}

;

sgn (σ) = sgn (σ^{- 1})

, so the row-sum and column-sum forms agree.

det (A B) = det (A) det (B) .

⟳fix

B

; the map

A \mapsto det (A B) / det (B)

(when

det B \neq = 0

) is multilinear alternating with value

1

I

, hence equals

det (A)

; the singular case follows by continuity/rank.

A invertible ⟺ det (A) \neq = 0, det (A^{- 1}) = \frac{1}{det ( A )} .

⟳from [4],

det (A) det (A^{- 1}) = det (I) = 1

; singularity means rows are dependent, forcing

det = 0

by alternation.

det (λ A) = λ^{n} det (A), λ \in F .

⟳pull a factor

λ

out of each of the

n

rows by multilinearity.

Cofactor (Laplace) expansion

Define the minor $M_{ij}$ as the determinant of the $(n - 1) \times (n - 1)$ matrix obtained by deleting row $i$ and column $j$ , and the cofactor $C_{ij} = (- 1)^{i + j} M_{ij}$ . Then for any fixed row $i$ (or column $j$ ):

det (A) = j = 1 \sum n a_{ij} C_{ij} = i = 1 \sum n a_{ij} C_{ij} .

⟳group the

n!

terms of [1] by which column the entry from row

i

occupies; the remaining factors form the minor with the correct sign

(- 1)^{i + j}

Moreover the adjugate $adj (A) = (C_{j i})$ (transpose of cofactors) satisfies

A \cdot adj (A) = adj (A) \cdot A = det (A) I, A^{- 1} = \frac{1}{det ( A )} adj (A) .

⟳the

(i, k)

entry of

A adj (A)

\sum_{j} a_{ij} C_{k j}

, which is

det (A)

i = k

and

0

otherwise (it equals the determinant of a matrix with a repeated row).

Why It Holds / Derivation

Row operations from the axioms

We derive the computational rules from axioms (i)–(iii).

Step 1 — Swapping two rows negates the determinant. Apply alternation to a matrix whose rows

i, k

are both set to

r_{i} + r_{k}

0 = D (\dots, r_{i} + r_{k}, \dots, r_{i} + r_{k}, \dots) .

⟳expand by multilinearity into four terms; the two with repeated rows vanish, leaving

D (\dots r_{i} \dots r_{k} \dots) + D (\dots r_{k} \dots r_{i} \dots) = 0

Step 2 — Adding a multiple of one row to another leaves the determinant unchanged. By multilinearity,

D (\dots, r_{i} + c r_{k}, \dots, r_{k}, \dots) = D (\dots, r_{i}, \dots, r_{k}, \dots) + c D (\dots, r_{k}, \dots, r_{k}, \dots),

and the last term is

0

by alternation.

⟳the added copy of

r_{k}

produces a matrix with two equal rows.

Step 3 — Triangular determinants. For an upper- (or lower-) triangular matrix

T

, in [1] only the identity permutation gives a nonzero product (any other forces a below-diagonal factor), so

det (T) = i = 1 \prod n t_{ii} .

⟳a nonidentity

σ

must map some

i

σ (i) < i

, hitting a zero entry above/below the diagonal.

Step 4 — Computation. Reduce

A

to upper-triangular

U

by row operations, tracking a sign factor

s = (- 1)^{# swaps}

and the product

p

of any scalings applied. Then

det (A) = s \cdot p^{- 1} \cdot \prod_{i} u_{ii}

. Pure elimination (Steps 1–2) needs no scaling, giving

O (n^{3})

cost.

■

Augustin-Louis Cauchy | 1789–1857

Cauchy gave the first systematic theory of determinants in his 1812 memoir, introducing the modern term déterminant in its current sense, proving the multiplication theorem $det (A B) = det (A) det (B)$ , and organizing the subject around what we now call the Leibniz expansion. He connected determinants to the theory of substitutions (permutations), anticipating group-theoretic signatures, and applied them to systems of equations and to the change-of-variables Jacobian in multiple integrals.

Building Intuition

Picture the rows of $A$ as edge vectors of a box (parallelepiped) anchored at the origin in $R^{n}$ . The determinant is the signed volume of that box.

If two edges point the same way, the box is flattened — volume $0$ — explaining the alternating axiom.
Stretching one edge by $α$ stretches the box's volume by $α$ — that's multilinearity.
Adding a multiple of one edge to another shears the box; a shear preserves volume — that's why row replacement doesn't change the determinant.
The sign records orientation: a negative determinant means the linear map flips handedness (a mirror reflection).

Multiplicativity $det (A B) = det (A) det (B)$ then says: applying two transformations in sequence multiplies their volume-scaling factors. A map that collapses volume to zero ( $det = 0$ ) cannot be undone — hence non-invertibility.

Determinant as signed area of the parallelogram spanned by two rows

Worked Examples

3×3 by cofactor expansion

Given:

A = 215 - 1 30 042 .

⟳choose row 1, which has a zero to reduce work.

Step 1 — Expand along row 1 using [7]:

det (A) = 2 C_{11} + (- 1) C_{12} + 0 \cdot C_{13} .

Step 2 — Cofactors:

C_{11} = + 3042 = 3 \cdot 2 - 4 \cdot 0 = 6,

C_{12} = - 1542 = - (1 \cdot 2 - 4 \cdot 5) = - (2 - 20) = 18.

Step 3 — Combine:

det (A) = 2 (6) + (- 1) (18) + 0 = 12 - 18 = - 6.

det (A) = - 6 (dimensionless)

Since $det (A) = - 6 \neq = 0$ , $A$ is invertible and the map flips orientation.

4×4 by row reduction

Given:

B = 1201241001121023 .

⟳use Step 4 of the proof — reduce to triangular form, tracking swaps and the diagonal product.

Step 1 — Eliminate column 1.

R_{2} \leftarrow R_{2} - 2 R_{1}

R_{4} \leftarrow R_{4} - R_{1}

(replacements,

det

unchanged):

1000 201 - 2 0112 1 - 2 22 .

Step 2 — Swap $R_{2} \leftrightarrow R_{3}$ to get a pivot in position

(2, 2)

. One swap multiplies

det

- 1

; track

s = - 1

1000 210 - 2 0112 12 - 2 2 .

Step 3 — Clear column 2.

R_{4} \leftarrow R_{4} + 2 R_{2}

100021000114 12 - 2 6 .

Step 4 — Clear column 3.

R_{4} \leftarrow R_{4} - 4 R_{3}

100021000110 12 - 2 14 .

Step 5 — Apply [11] with sign. Diagonal product

= 1 \cdot 1 \cdot 1 \cdot 14 = 14

; include the swap sign

s = - 1

det (B) = s \cdot 14 = - 14.

det (B) = - 14 (dimensionless)

Common Misconceptions

The determinant is not additive in the matrix. In general

det (A + B) \neq = det (A) + det (B) .

⟳multilinearity is row-by-row, not matrix-wide; adding entire matrices mixes all rows simultaneously.

Other frequent errors:

$det (λ A) = λ^{n} det (A)$ , not $λ det (A)$ (see [6]).
A row swap flips the sign; forgetting this in elimination gives the wrong sign.
The cofactor sign is $(- 1)^{i + j}$ from the position, not from the entry's own sign.
The adjugate is the transpose of the cofactor matrix; using the untransposed version gives a wrong inverse for non-symmetric $A$ .

Let

A = (1001), B = (- 1 0 0 - 1) .

Then

det (A) = 1

det (B) = 1

, but

A + B = 0

with

det (A + B) = 0 \neq = 1 + 1 = 2

. This decisively refutes additivity. Note also

B = - I

and

det (- I) = (- 1)^{2} det (I) = 1

, confirming [6] with

λ = - 1, n = 2

— not

- 1

Connections

The determinant ties together many strands of linear algebra:

It equals the product of the Eigenvalues (with algebraic multiplicity): $det (A) = \prod_{i} λ_{i}$ , the constant term (up to sign) of the Characteristic Polynomial.
It governs solvability in Cramer's Rule: $x_{i} = det (A_{i}) / det (A)$ for $A x = b$ .
It is the top exterior power in the language of Exterior Algebra: $det (A)$ is the scalar by which $A$ acts on $Λ^{n} V$ .
In multivariable calculus it appears as the Jacobian Determinant controlling the change-of-variables factor in integration.
Its sign defines Orientation (Linear Algebra), and its magnitude the volume distortion measured by Volume and Measure.
For $2 \times 2$ and $3 \times 3$ matrices it underlies the Cross Product and the scalar triple product.

Real-World Applications

Computer graphics & geometry: The signed area/volume via determinants determines triangle orientation (back-face culling), point-in-triangle tests, and the $3 \times 3$ Jacobian for shading normals.
Robotics & kinematics: The Jacobian determinant of a manipulator detects singular configurations where the robot loses a degree of freedom (det $\to 0$ ).
Differential equations: The Wronskian (a determinant of solutions and their derivatives) tests linear independence of solutions of an ODE; nonzero $\Rightarrow$ a fundamental set.
Statistics: The determinant of a covariance matrix (the generalized variance) appears in the multivariate Gaussian density and in measuring data spread/collinearity.
Physics — quantum mechanics: Slater determinants build antisymmetric many-fermion wavefunctions, automatically enforcing the Pauli exclusion principle.
Numerical analysis & cryptography: Determinants govern matrix invertibility in solving linear systems, and integer determinants (e.g. of lattice bases) underlie lattice-based methods and the Smith normal form.
Economics (input–output models): Determinant conditions (Hawkins–Simon) guarantee a viable, nonnegative solution to Leontief production systems.

Historical Development

1683 — Seki Takakazu (Japan) introduces determinant-like elimination methods for systems of equations, independently of Europe.
1693 — Gottfried Leibniz describes, in a letter to L'Hôpital, the alternating-sum rule for solving linear systems — essentially formula [1].
1750 — Gabriel Cramer publishes Cramer's Rule, giving determinant formulas for solutions of linear systems.
1772 — Pierre-Simon Laplace & Alexandre-Théophile Vandermonde develop cofactor (Laplace) expansion and structured determinants.
1812 — Augustin-Louis Cauchy delivers a unified theory, proves the multiplication theorem $det (A B) = det (A) det (B)$ , and standardizes the term determinant.
1841 — Carl Gustav Jacob Jacobi systematizes the functional ("Jacobian") determinant for change of variables in his treatise De determinantibus functionalibus.
1850s — James Joseph Sylvester & Arthur Cayley introduce matrix notation and connect determinants to the emerging theory of matrices and invariants.
1858 — Cayley states the Cayley–Hamilton theorem, linking $det$ , the characteristic polynomial, and matrix powers.
20th c. — Emil Artin & Nicolas Bourbaki recast determinants axiomatically as the unique normalized alternating multilinear form, the modern viewpoint of equation [2].

Subtleties & Limits

Computational and theoretical frontiers.

Algebraic complexity: Determinant lies in the complexity class $VP$ (efficiently computable), whereas the superficially similar permanent (same formula without signs) is $# P$ -complete (Valiant, 1979). The Determinant vs. Permanent problem — whether the permanent of an $n \times n$ matrix can be expressed as the determinant of a polynomially larger matrix — is a central open question in geometric complexity theory.
Numerical stability: The determinant is a poor invertibility test in floating point; a tiny determinant may reflect scaling rather than singularity. Practitioners prefer the condition number or rank-revealing factorizations. Computing $det$ itself can overflow/underflow; one works with $lo g ∣ det ∣$ .
Beyond fields: Over a commutative ring (e.g. $Z$ , polynomial rings), $det$ is defined and invertibility requires $det$ to be a unit — not merely nonzero — connecting to the Smith normal form and module theory.
Infinite dimensions: Generalizing $det$ to operators on Hilbert spaces requires the theory of Fredholm and trace-class determinants, with deep analytic subtleties.