The determinant is the unique normalized alternating multilinear function of a square matrix's columns. We construct it over any commutative ring, prove its main properties, and explain its meaning as signed volume over \(\mathbb R\).
Fix \(n\geq1\) and a commutative ring \(R\) with identity. Write \(M_n(R)\) for the \(n\times n\) matrices over \(R\), \(I_n\) for the identity, and \(e_1,\dots,e_n\) for the standard basis of \(R^n\). A matrix \(A=(a_{ij})\) is an ordered list of columns:
\[ A=[v_1\ \cdots\ v_n],\qquad v_j=\sum_{i=1}^n a_{ij}e_i. \]The construction applies to fields, \(\mathbb Z\), \(\mathbb Z/m\mathbb Z\), and polynomial rings; it requires no division. For computation over a field, see Gaussian elimination.
1. Alternating multilinear maps
Definition 1.1. Multilinear, alternating, normalized, and antisymmetric
A function \(F:(R^n)^n\to R\) is:
- Multilinear if it is \(R\)-linear in each slot with the others fixed: \[F(\dots,ru+sv,\dots)=rF(\dots,u,\dots)+sF(\dots,v,\dots).\]
- Alternating if it vanishes whenever two arguments are equal.
- Normalized if \(F(e_1,\dots,e_n)=1\).
- Antisymmetric if exchanging two arguments negates its value.
Lemma 1.2. Alternation implies antisymmetry
Every alternating multilinear function is antisymmetric, over any commutative ring.
Proof.
Put \(u+v\) in both selected slots. Alternation and multilinearity give
\[ 0=F(\dots,u+v,\dots,u+v,\dots) =F(\dots,u,\dots,v,\dots)+F(\dots,v,\dots,u,\dots), \]since the two repeated-argument terms vanish. No division by \(2\) is needed.
□Example 1.3. Antisymmetry need not imply alternation
Over \(\mathbb F_2=\mathbb Z/2\mathbb Z\), the bilinear function \(F(u,v)=u_1v_1\), for \(u,v\in\mathbb F_2^2\), is antisymmetric: \(F(v,u)=F(u,v)=-F(u,v)\). But \(F(e_1,e_1)=1\), so it is not alternating.
In general, antisymmetry on equal arguments gives only \(2F(\dots,u,\dots,u,\dots)=0\). If \(2\) is a unit in \(R\), this does imply alternation.
2. The sign of a permutation
Let \(S_n\) be the permutations of \(\{1,\dots,n\}\), with \((\sigma\rho)(i)=\sigma(\rho(i))\). A transposition \((p\ q)\) exchanges \(p\) and \(q\), fixing the other indices.
Definition 2.1. Inversions and sign
An inversion of \(\sigma\in S_n\) is a pair \(i<j\) with \(\sigma(i)>\sigma(j)\). Define
\[ \operatorname{inv}(\sigma)=\#\{(i,j):i<j,\ \sigma(i)>\sigma(j)\}, \qquad \operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}. \]The sign is an integer \(\pm1\), interpreted as \(\pm1_R\) in ring-valued formulas.
Theorem 2.2. Sign and transposition parity
The sign is a homomorphism \(S_n\to\{+1,-1\}\), and every transposition has sign \(-1\). Every permutation is a product of transpositions; if \(\sigma=\tau_1\cdots\tau_m\), then
\[\operatorname{sgn}(\sigma)=(-1)^m.\]Thus the parity of a transposition factorization is independent of the factorization.
Proof.
Right composition with an adjacent transposition exchanges two adjacent values in the one-line list \([\sigma(1),\dots,\sigma(n)]\). Their mutual inversion toggles; comparisons with every other position have the same total contribution. The sign therefore flips.
For \(p<q\), put \(s_i=(i\ i+1)\). The factorization
\[ (p\ q)=s_p s_{p+1}\cdots s_{q-2}s_{q-1}s_{q-2}\cdots s_p \]has \(2(q-p)-1\) factors, so every transposition flips the sign. Sorting a permutation by adjacent swaps shows that transpositions generate \(S_n\). Starting from the identity, any factorization with \(m\) transpositions gives sign \((-1)^m\). Concatenating factorizations of \(\sigma\) and \(\rho\) now gives \(\operatorname{sgn}(\sigma\rho)=\operatorname{sgn}(\sigma)\operatorname{sgn}(\rho)\).
□Two further descriptions of sign
Proposition 2.3. The Vandermonde description
For \(\Delta=\prod_{i<j}(x_j-x_i)\in\mathbb Z[x_1,\dots,x_n]\),
\[\prod_{i<j}(x_{\sigma(j)}-x_{\sigma(i)})=\operatorname{sgn}(\sigma)\Delta.\]Proof.
Permuting indices permutes the unordered pairs in the product. A factor changes sign exactly when \(\sigma(i)>\sigma(j)\). There is one such change per inversion.
□Proposition 2.4. Uniqueness of the sign homomorphism
For \(n\geq2\), sign is the only nontrivial homomorphism \(S_n\to\{+1,-1\}\).
Proof.
All transpositions are conjugate: \(\rho(p\ q)\rho^{-1}=(\rho(p)\ \rho(q))\). A homomorphism to the abelian group \(\{\pm1\}\) therefore has the same value on every transposition. Since they generate \(S_n\), the value \(+1\) gives the trivial homomorphism and \(-1\) gives sign.
□3. Construction and uniqueness
Definition 3.1. The determinant
For \(A\in M_n(R)\), define the determinant by the Leibniz formula:
\[ \boxed{\det A=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma) \prod_{j=1}^n a_{\sigma(j),j}.} \]Each of the \(n!\) summands chooses one entry from every row and every column. For later minor formulas, set the determinant of the empty \(0\times0\) matrix to \(1\).
For example,
\[ \det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc, \]and the six permutations of \(S_3\) give
\[ \begin{aligned} \det A={}&a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}\\ &-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}. \end{aligned} \]Theorem 3.2. Existence
The Leibniz determinant is multilinear, alternating, and normalized over every commutative ring.
Proof.
Each monomial contains exactly one entry from each column, so it is linear in that column. At \(I_n\), only the identity permutation contributes, giving \(\det I_n=1\).
If columns \(p\) and \(q\) are equal, pair each \(\sigma\) with \(\sigma\circ(p\ q)\). This is a fixed-point-free involution on \(S_n\). The paired signs are opposite, while the products agree: equality of columns exchanges the two selected entries, and commutativity lets us reorder the factors. Every pair cancels, so \(\det A=0\). This proves alternation even in characteristic \(2\).
□Lemma 3.3. Rigidity
If \(F:(R^n)^n\to R\) is alternating and multilinear, then
\[F(A)=F(I_n)\det A.\]Proof.
Expand the columns in the standard basis:
\[ F(A)=\sum_{k_1,\dots,k_n=1}^n \left(\prod_{j=1}^n a_{k_j,j}\right)F(e_{k_1},\dots,e_{k_n}). \]Repeated indices give zero by alternation. The remaining tuples are \(k_j=\sigma(j)\), and Lemma 1.2 with Theorem 2.2 gives
\[ F(e_{\sigma(1)},\dots,e_{\sigma(n)}) =\operatorname{sgn}(\sigma)F(e_1,\dots,e_n). \]Substitution gives the claimed formula.
□Theorem 3.4. Uniqueness
The determinant is the unique normalized alternating multilinear function \((R^n)^n\to R\).
Proof.
Theorem 3.2 gives existence. If \(F\) has the three properties, Lemma 3.3 and \(F(I_n)=1\) give \(F(A)=\det A\).
□4. Fundamental properties
Proposition 4.1. Transpose and row symmetry
For every \(A\in M_n(R)\), \(\det(A^{\mathsf T})=\det A\). Consequently, the determinant is alternating and multilinear in the rows as well as the columns.
Proof.
Reindex the Leibniz formula by \(\rho=\sigma^{-1}\):
\[ \det(A^{\mathsf T}) =\sum_\sigma\operatorname{sgn}(\sigma)\prod_j a_{j,\sigma(j)} =\sum_\rho\operatorname{sgn}(\rho)\prod_i a_{\rho(i),i} =\det A. \]Here commutativity allows product reordering, and \(\operatorname{sgn}(\sigma^{-1})=\operatorname{sgn}(\sigma)\). Transposition converts row operations into column operations.
□Proposition 4.2. Row and column transformations
For either rows or columns:
- Exchanging two multiplies the determinant by \(-1\).
- Scaling one by \(c\in R\) multiplies the determinant by \(c\).
- Adding a multiple of one to a different one leaves the determinant unchanged.
A zero or repeated row or column therefore gives determinant zero.
Proof.
The first two assertions are antisymmetry and multilinearity. For replacement, expanding in the changed slot gives the original determinant plus a multiple of one with two equal arguments; the latter vanishes. A zero argument gives zero by linearity.
□Scaling by any \(c\) obeys the determinant formula, but is a reversible row operation only when \(c\) is a unit.
Proposition 4.3. Triangular matrices
If \(A\) is upper triangular (\(a_{ij}=0\) for \(i>j\)) or lower triangular (\(a_{ij}=0\) for \(i<j\)), then
\[\det A=\prod_{i=1}^n a_{ii}.\]Proof.
For upper triangular \(A\), a nonzero Leibniz term requires \(\sigma(j)\leq j\) for every \(j\). Since \(\sum_j\sigma(j)=\sum_j j\), every inequality must be equality. Only the identity term remains. The lower triangular case follows by transposition.
□Over a field, Gaussian elimination reduces a matrix to triangular form. Track the row swaps and scalings, then multiply the diagonal entries. This takes \(O(n^3)\) arithmetic operations rather than evaluating the \(n!\)-term sum.
Theorem 4.4. Multiplicativity
For all \(A,B\in M_n(R)\), \(\det(AB)=\det A\det B\).
Proof.
Fix \(A\) and set \(G(B)=\det(AB)\). Each column of \(AB\) is \(A\) times the corresponding column of \(B\), so \(G\) is multilinear. Equal columns of \(B\) give equal columns of \(AB\), so \(G\) is alternating. Lemma 3.3 yields
\[\det(AB)=G(I_n)\det B=\det A\det B.\] □Corollary 4.5. Products, inverses, and similarity
For \(m\geq0\), \(\det(A^m)=(\det A)^m\). If \(A\) is invertible, \(\det(A^{-1})=(\det A)^{-1}\). For invertible \(P\),
\[\det(P^{-1}AP)=\det A.\]Proof.
Apply multiplicativity repeatedly. In particular, \(1=\det(AA^{-1})=\det A\det(A^{-1})\), and the factors \(\det P\) and \(\det(P^{-1})\) cancel in the similarity formula.
□5. Cofactors and invertibility
Definition 5.1. Minors, cofactors, and adjugate
Delete row \(i\) and column \(j\) of \(A\) to obtain \(A^{(i,j)}\). The corresponding minor is \(\det A^{(i,j)}\), the cofactor is
\[C_{ij}=(-1)^{i+j}\det A^{(i,j)},\]and the adjugate is the transpose of the cofactor matrix: \(\operatorname{adj}(A)_{ij}=C_{ji}\). Our convention \(\det(\varnothing)=1\) includes \(n=1\).
Proposition 5.2. Cofactor expansion
Along any row \(i\) or column \(j\),
\[\det A=\sum_{k=1}^n a_{ik}C_{ik}=\sum_{k=1}^n a_{kj}C_{kj}.\]Proof.
Group the Leibniz terms according to the selected entry \(a_{kj}\) in column \(j\). Move row \(k\) and column \(j\) to the first positions, using \(k-1+j-1\) swaps. The remaining choices are the permutations of the minor, so this group sums to \(a_{kj}(-1)^{k+j}\det A^{(k,j)}\). Summing over \(k\) gives the column formula; transposition gives the row formula.
□Theorem 5.3. Adjugate identity
Proof.
The \((i,j)\) entry of \(A\operatorname{adj}(A)\) is \(\sum_k a_{ik}C_{jk}\). For \(i=j\), cofactor expansion gives \(\det A\). Otherwise this is the row-\(j\) expansion after replacing row \(j\) by row \(i\). The cofactors are unchanged, and the repeated rows make the determinant zero. The column argument gives \(\operatorname{adj}(A)A\).
□Theorem 5.4. Invertibility criterion
A matrix \(A\in M_n(R)\) is invertible exactly when \(\det A\) is a unit, meaning it has a multiplicative inverse in \(R\). In that case,
\[A^{-1}=(\det A)^{-1}\operatorname{adj}(A).\]Over a field, this is equivalent to \(\det A\neq0\). Over \(\mathbb Z\), it means \(\det A=\pm1\).
Proof.
If \(A\) is invertible, multiplicativity gives \(\det A\det(A^{-1})=1\). Conversely, if \(\det A\) is a unit, multiplying the adjugate identity by \((\det A)^{-1}\) gives the stated two-sided inverse. The only integer units are \(\pm1\).
□Corollary 5.5. Cramer's rule
Let \(A_j(b)\) be \(A\) with column \(j\) replaced by \(b\). Every solution of \(Ax=b\) satisfies
\[\det A\,x_j=\det A_j(b).\]If \(\det A\) is a unit, the solution is unique and \(x_j=(\det A)^{-1}\det A_j(b)\).
Proof.
Write \(b=\sum_k x_kv_k\) in terms of the columns of \(A\) and expand \(\det A_j(b)\) by multilinearity. Every term with \(k\neq j\) has repeated columns and vanishes; the remaining term is \(x_j\det A\). A unit determinant makes \(A\) invertible, giving uniqueness and allowing division.
□6. Why commutativity matters
Example 6.1. Failure of the ordered Leibniz formula
Over the quaternions, take products in increasing column order. The matrix
\[ A=\begin{bmatrix}i&i\\j&j\end{bmatrix} \quad\text{gives}\quad a_{11}a_{22}-a_{21}a_{12}=ij-ji=2k\neq0, \]using \(ij=k\) and \(ji=-k\). Its columns are equal, so this formula is not alternating. The cancellation proof of Theorem 3.2 requires commutativity to make paired monomials agree.
7. Signed volume over the reals
Theorem 7.1. Volume of a parallelepiped
For a real matrix \(A=[v_1\ \cdots\ v_n]\), let
\[P_A=\left\{\sum_{j=1}^n t_jv_j:0\leq t_j\leq1\right\}.\]Its \(n\)-dimensional volume is \(\lvert\det A\rvert\), including zero volume when \(A\) is singular.
Proof sketch.
For invertible \(A\), elimination factors the linear map into coordinate swaps, nonzero coordinate scalings, and shears. Swaps preserve volume, a scaling by \(c\) multiplies volume by \(|c|\), and a shear preserves volume by translating one-dimensional slices and applying Fubini's theorem. These are exactly the absolute determinant factors in Proposition 4.2. Apply them to the unit cube, whose volume is \(1\).
If \(A\) is singular, \(P_A\) lies in a proper linear subspace, which has \(n\)-dimensional measure zero; Theorem 5.4 gives \(\det A=0\).
□Definition 7.2. Orientation and signed volume
Relative to the standard ordered basis, a basis with column matrix \(A\) is positively oriented when \(\det A>0\), and negatively oriented when \(\det A<0\). More generally, two ordered bases have the same orientation when their change-of-basis matrix has positive determinant. Multiplicativity makes this an equivalence relation.
The signed volume is the ordinary volume with this orientation sign, hence equals \(\det A\). For a singular frame the volume is zero; no orientation sign is assigned.
For \(A=\begin{bmatrix}2&1\\1&2\end{bmatrix}\), the area is \(3\) and the orientation is positive. Swapping columns gives \(A'=\begin{bmatrix}1&2\\2&1\end{bmatrix}\), with the same area and determinant \(-3\).
The animations interpolate from \(I_2\) to these matrices. The first has determinant \(1+2t\); the second has determinant \(1-4t^2\), collapsing at \(t=\tfrac12\) before reversing orientation.
Sources and further reading
Item references accompany the statements above. The main Alphabeta Math chapters are Determinants of Matrices over a Commutative Ring and Symmetric Groups and the Sign Homomorphism. They use indices starting at zero; this article uses indices starting at one. The Vandermonde product here uses \(x_j-x_i\), reversing the factors in the linked definition without changing its permutation law.
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