Determinants — Properties and Computation

⚡ Key Takeaways

The determinant is a scalar $\det(A)$ assigned to every square matrix $A$ , encoding whether $A$ is invertible (nonzero $\iff$ invertible), the signed volume scaling of the linear map $A$ , and the orientation it induces. Key results: multiplicativity $\det(AB)=\det(A)\det(B)$ , 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 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.

DefinitionDeterminant via the Leibniz formula

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

1

\quad \det(A)=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}a_{i,\sigma(i)},

where $S_n$ is the symmetric group of all permutations of $\{1,\dots,n\}$ , and $\operatorname{sgn}(\sigma)\in\{+1,-1\}$ is the signature of $\sigma$ (equal to $(-1)^{k}$ where $k$ is the number of transpositions composing $\sigma$ ). Derivation hint: 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.

DefinitionDeterminant via axioms (Weierstrass / Artin)

The determinant is the unique function $D:M_n(\mathbb{F})\to\mathbb{F}$ satisfying, with $A$ written by its rows $r_1,\dots,r_n$ : (i) Multilinearity: linear in each row separately,

2

\quad D(\dots,\alpha r_i+\beta r_i',\dots)=\alpha\,D(\dots,r_i,\dots)+\beta\,D(\dots,r_i',\dots).

Derivation hint: 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].

TheoremFundamental properties

For $A,B\in M_n(\mathbb{F})$ :

3

\quad \det(A^{\mathsf T})=\det(A).

Derivation hint: in [1], reindex by $\tau=\sigma^{-1}$ ; $\operatorname{sgn}(\sigma)=\operatorname{sgn}(\sigma^{-1})$ , so the row-sum and column-sum forms agree.

4

\quad \det(AB)=\det(A)\det(B).

Derivation hint: fix $B$ ; the map $A\mapsto\det(AB)/\det(B)$ (when $\det B\ne 0$ ) is multilinear alternating with value $1$ at $I$ , hence equals $\det(A)$ ; the singular case follows by continuity/rank.

5

\quad A \text{ invertible} \iff \det(A)\neq 0,\qquad \det(A^{-1})=\frac{1}{\det(A)}.

Derivation hint: from [4], $\det(A)\det(A^{-1})=\det(I)=1$ ; singularity means rows are dependent, forcing $\det=0$ by alternation.

6

\quad \det(\lambda A)=\lambda^{n}\det(A),\qquad \lambda\in\mathbb{F}.

Derivation hint: pull a factor $\lambda$ out of each of the $n$ rows by multilinearity.

TheoremCofactor (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 $j$ ):

7

\quad \det(A)=\sum_{j=1}^{n}a_{ij}\,C_{ij}=\sum_{i=1}^{n}a_{ij}\,C_{ij}.

Derivation hint: 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 $\operatorname{adj}(A)=(C_{ji})$ (transpose of cofactors) satisfies

8

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

Derivation hint: the $(i,k)$ entry of $A\,\operatorname{adj}(A)$ is $\sum_j a_{ij}C_{kj}$ , which is $\det(A)$ if $i=k$ and $0$ otherwise (it equals the determinant of a matrix with a repeated row).

ProofRow 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 $r_i+r_k$ :

9

\quad 0=D(\dots,r_i+r_k,\dots,r_i+r_k,\dots).

Derivation hint: 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,

10

\quad 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. Derivation hint: 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

11

\quad \det(T)=\prod_{i=1}^{n}t_{ii}.

Derivation hint: a nonidentity $\sigma$ must map some $i$ to $\sigma(i)<i$ , hitting a zero entry above/below the diagonal.

Step 4 — Computation. Reduce $A$ to upper-triangular $U$ by row operations, tracking a sign factor $s=(-1)^{\#\text{swaps}}$ 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. $\qquad\blacksquare$

∎

mathematician · 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 $\det(AB)=\det(A)\det(B)$ , 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.

🧠 Intuition

Picture the rows of $A$ as edge vectors of a box (parallelepiped) anchored at the origin in $\mathbb{R}^n$ . 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 $\alpha$ stretches the box's volume by $\alpha$ — 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 $\det(AB)=\det(A)\det(B)$ 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.

Figure

Determinant as signed area of the parallelogram spanned by two rows

Example3×3 by cofactor expansion

Given:

12

\quad A=\begin{pmatrix}2&-1&0\\ 1&3&4\\ 5&0&2\end{pmatrix}.

Derivation hint: 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}=+\begin{vmatrix}3&4\\0&2\end{vmatrix}=3\cdot2-4\cdot0=6,

C_{12}=-\begin{vmatrix}1&4\\5&2\end{vmatrix}=-(1\cdot2-4\cdot5)=-(2-20)=18.

Step 3 — Combine:

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

\boxed{\det(A)=-6}\quad(\text{dimensionless})

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

Example4×4 by row reduction

Given:

13

\quad B=\begin{pmatrix}1&2&0&1\\ 2&4&1&0\\ 0&1&1&2\\ 1&0&2&3\end{pmatrix}.

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

Step 1 — Eliminate column 1. $R_2\leftarrow R_2-2R_1$ , $R_4\leftarrow R_4-R_1$ (replacements, $\det$ unchanged):

\begin{pmatrix}1&2&0&1\\ 0&0&1&-2\\ 0&1&1&2\\ 0&-2&2&2\end{pmatrix}.

Step 2 — Swap $R_2\leftrightarrow R_3$ to get a pivot in position $(2,2)$ . One swap multiplies $\det$ by $-1$ ; track $s=-1$ :

\begin{pmatrix}1&2&0&1\\ 0&1&1&2\\ 0&0&1&-2\\ 0&-2&2&2\end{pmatrix}.

Step 3 — Clear column 2. $R_4\leftarrow R_4+2R_2$ :

\begin{pmatrix}1&2&0&1\\ 0&1&1&2\\ 0&0&1&-2\\ 0&0&4&6\end{pmatrix}.

Step 4 — Clear column 3. $R_4\leftarrow R_4-4R_3$ :

\begin{pmatrix}1&2&0&1\\ 0&1&1&2\\ 0&0&1&-2\\ 0&0&0&14\end{pmatrix}.

Step 5 — Apply [11] with sign. Diagonal product $=1\cdot1\cdot1\cdot14=14$ ; include the swap sign $s=-1$ :

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

\boxed{\det(B)=-14}\quad(\text{dimensionless})

⚠️ Common mistake

The determinant is not additive in the matrix. In general

14

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

Derivation hint: multilinearity is row-by-row, not matrix-wide; adding entire matrices mixes all rows simultaneously. Other frequent errors:

$\det(\lambda A)=\lambda^n\det(A)$ , not $\lambda\det(A)$ (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 $A$ .

Counterexample

Let

A=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad B=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}.

Then $\det(A)=1$ , $\det(B)=1$ , but $A+B=0$ with $\det(A+B)=0\ne 1+1=2$ . This decisively refutes additivity. Note also $B=-I$ and $\det(-I)=(-1)^2\det(I)=1$ , confirming [6] with $\lambda=-1,n=2$ — not $-1$ .

connection

The determinant ties together many strands of linear algebra:

It equals the product of the Eigenvalues (with algebraic multiplicity): $\det(A)=\prod_i \lambda_i$ , 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 $Ax=b$ .
It is the top exterior power in the language of Exterior Algebra: $\det(A)$ is the scalar by which $A$ acts on $\Lambda^n V$ .
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 $2\times2$ and $3\times3$ matrices it underlies the Cross Product and the scalar triple product.

🔬 Where this appears

⏳ History

- 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(AB)=\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].

open-problem

Computational and theoretical frontiers.

Algebraic complexity: Determinant lies in the complexity class $\mathsf{VP}$ (efficiently computable), whereas the superficially similar permanent (same formula without signs) is $\#\mathsf P$ -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 $\log|\det|$ .
Beyond fields: Over a commutative ring (e.g. $\mathbb{Z}$ , 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.

Determinants — Properties and Computation

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also