Determinants: Linear Algebra Study Notes

🔢 Determinants

• Definition and notation of the determinant
• Low dimensions: empty, 1 × 1, 2 × 2 (parallelogram area) and 3 × 3 (rule of Sarrus)
• Equivalent definitions: multiplicative property, elementary transformations, Leibniz formula, alternating multilinear form
• Determinants over a commutative ring and generic determinants
• Properties: zero test, transformations, invertibility, Laplace expansion, adjugate matrix
• Invariance under ring homomorphisms and the Hadamard bound
• Other determinants: endomorphisms, n vectors, quadratic forms
• Derivative of the determinant (Jacobi's formula)
• Computation: Gaussian elimination, decompositions, faster methods
• Uses in linear algebra, geometry, eigenvalues, functions (Jacobian, Wronskian) and vector geometry
• Generalizations: immanant, exterior algebra, finite-dimensional algebras

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💡 What Is a Determinant?

The determinant is a scalar-valued function of the entries of a square matrix. It has many properties that make it fundamental for the study of square matrices and of the linear transformations they represent.

• Common notations for the determinant of a matrix $A$: $\det(A)$, $\det A$, or $|A|$.
• For dimensions two and three, it is explicitly:

$\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc,$

$\det\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=aei+bfg+cdh-ceg-bdi-afh.$

Key ideas at a glance

• A simple, conceptual definition: the determinant is the unique function that maps a product of matrices to a product of scalars and whose value on a triangular matrix is the product of the diagonal entries.
• So the determinant of a product of two square matrices is the product of their determinants.
• A square matrix is invertible if and only if its determinant has a multiplicative inverse. Over a field, this means the determinant is nonzero.
• The adjugate matrix gives an explicit expression of the inverse matrix.
• Two similar matrices have the same determinant, so the determinant of a linear transformation is well defined.
• For real linear transformations, the determinant is the scale factor by which the transformation alters every volume:
  • positive if orientation is preserved
  • negative if orientation is reversed
  • zero if the transformation is not bijective
• The determinant is linear in each row and each column, so it is a multilinear function.
• Laplace expansion expresses the coefficients of this linear combination as determinants of one dimension less, called cofactors. It is useful when rows or columns are very sparse (only one or two nonzero entries).

Where determinants occur

Area	Role of the determinant
Linear systems	Cramer's rule expresses the solution in terms of determinants
Eigenvalues	The characteristic polynomial is a determinant with polynomial entries; its roots are the eigenvalues
Geometry	Areas, volumes, space orientation and colinearity
Analysis	The Jacobian determinant gives the change of variables in multiple integrals; its non-nullity is a hypothesis of the implicit function theorem
Algebraic geometry	The Hessian determinant allows finding inflexion points of an algebraic plane curve

Determinants can be computed efficiently with Gaussian elimination and singular value decomposition. The Leibniz formula or Laplace expansion can compute them theoretically, but this is inefficient except in special cases.

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📐 Low Dimensions

Dimension below 2

• Matrices with fewer than two rows and columns are rarely considered per se, but may occur when working with $n \times n$ matrices where $n$ is an unspecified natural number.
• A $0 \times 0$ matrix is the empty matrix (no rows, no columns, no entries). By the usual conventions for the empty set and the empty product, its determinant is 1.
• A $1 \times 1$ matrix has a single entry, and its determinant equals that entry: $\det[a]=a$.

Coherency of these conventions. For an $n\times n$ matrix $A$ with nonzero determinant $d$, the inverse is the multiplication by $1/d$ of the adjugate matrix, whose entries are (up to sign) determinants of $(n-1)\times(n-1)$ submatrices. For $n=1$, the single entry of the adjugate is the determinant of the empty matrix, which is 1. So

$[d]^{-1}=\tfrac{1}{d}\,[1]=\left[\tfrac{1}{d}\right],$

which is what is expected for $1\times 1$ matrices.

Two by two matrices

$\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc.$

This simple form allows verifying by direct computation the properties listed under Properties below.

Parallelogram area

A main use of $2\times 2$ determinants is computing the area of the parallelogram defined by two vectors in a Euclidean plane.

• The column vectors $\begin{pmatrix}a\\c\end{pmatrix}$ and $\begin{pmatrix}b\\d\end{pmatrix}$ define a parallelogram with vertices $(0,0)$, $(a,c)$, $(b,d)$ and $(a+b,c+d)$.
• Its area is the absolute value of $ad-bc$.
• The sign gives the relative orientation of the vectors:
  • positive if one passes from $(a,c)$ to $(b,d)$ by a counterclockwise rotation
  • negative if a clockwise rotation is needed
  • zero if the vectors are aligned
• So the determinant is the signed area of the parallelogram. The proof can be done with trigonometry or by dissection and rearrangement.

3 × 3 matrices

By the Leibniz formula:

$\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}=aei+bfg+cdh-ceg-bdi-afh.$

• The rule of Sarrus is a mnemonic for this expanded form: the sum of the products of the three diagonal lines running north-west to south-east, minus the sum of the products of the three diagonal lines running south-west to north-east, when copies of the first two columns are written beside the matrix.
• This scheme does not carry over into higher dimensions.

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📚 Definitions

The determinant of a square matrix is a scalar uniquely associated to it. Commonly the entries and determinants are real or complex numbers, but more generally the entries may belong to any commutative ring, in which case the determinant belongs to this ring.

For a square matrix with $n$ rows and columns

$A=\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots&a_{1,n}\\a_{2,1}&a_{2,2}&\cdots&a_{2,n}\\\vdots&\vdots&\ddots&\vdots\\a_{n,1}&a_{n,2}&\cdots&a_{n,n}\end{bmatrix},$

the determinant is denoted $\det(A)$, $\det A$, or by vertical bars around the array of entries.

• Several equivalent definitions exist; which is primary depends on the author and the level of generality.
• Here, the multiplicative property is chosen as the primary definition because it is simple and needs no specific background.
• In this section the scalars are assumed to belong to a field. The multiplicative and elementary-transformation characterizations remain valid over an integral domain, but their equivalence requires a different argument (fraction-free elimination and cancellation instead of ordinary Gaussian elimination).

Definition by multiplicative property

The determinant is the unique function from square matrices to scalars that

1. maps a product of matrices of the same size to the product of their determinants: $\det(AB)=\det(A)\det(B)$
2. maps every triangular matrix to the product of its diagonal entries: $a_{1,1}\cdots a_{n,n}$

Such a definition requires a proof that the function exists and is unique. This is a byproduct of the proof of equivalence with the other definitions.

Definition by elementary transformations

An elementary row transformation is one of three types, where $R_i$ denotes the $i$th row:

Type	Transformation
1	Multiply one row by a scalar: $R_i\to\lambda R_i$
2	Add to a row a scalar multiple of another row: $R_i\to R_i+\lambda R_j$
3	Exchange two rows: $(R_i,R_j)\to(R_j,R_i)$

• Elementary column transformations are defined similarly.
• Type 3 can be obtained as a succession of types 1 and 2:

$R_i\to R_i+R_j,\qquad R_j\to R_j-R_i,\qquad R_i\to R_i+R_j,\qquad R_j\to -R_j$

Definition. The determinant is the unique function of the entries of a square matrix such that:

• the determinant of the identity matrix is 1
• type 1 transformations multiply the determinant by the involved scalar
• type 2 transformations do not change the determinant

It follows that type 3 transformations multiply the determinant by $-1$.

Why it is equivalent to the multiplicative definition:

• $R_i\to\lambda R_i$ amounts to left-multiplying by a diagonal matrix with all diagonal entries 1 except the $i$th, which is $\lambda$.
• $R_i\to R_i+\lambda R_j$ amounts to left-multiplying by a triangular matrix with all diagonal entries 1, entry $\lambda$ in row $i$ and column $j$, and all other entries zero.
• A function satisfying the multiplicative definition satisfies the elementary one, since the identity matrix is triangular and the other conditions are the multiplicative property with an elementary matrix as left factor.
• Conversely, over a field Gaussian elimination shows every matrix is a product of elementary matrices (zero allowed as a scalar for type 1). To prove $\det(A)\det(B)=\det(AB)$, express $A$ as such a product and use associativity of the matrix product.
• Uniqueness also follows from Gaussian elimination: any matrix can be transformed to a row echelon form (a triangular matrix). This shows that two determinant functions agree on every such path and are therefore identical. It does not by itself prove existence, since it does not show the value is independent of the elimination path. A function with the defining properties must still be constructed.

Leibniz formula

The Leibniz formula is one of the most common definitions. It is an explicit expression of the determinant as a multivariate polynomial in the matrix entries, and it makes sense over a general commutative ring (no field or integral domain needed). Proving it satisfies the preceding definitions gives the required existence proof.

It uses permutations and their signatures:

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

where $\sigma$ runs over all permutations of $\{1,\ldots,n\}$.

• For the empty matrix, $\det[~]=1$. For $n=1$, $\det[a_{1,1}]=a_{1,1}$.
• Otherwise the formula is often expanded as

$\det(A)=\sum_{\sigma}\operatorname{sgn}(\sigma)\,a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}.$

Levi-Civita symbol variant. The symbol $\varepsilon_{i_1,\ldots,i_n}$ is defined on $n$-tuples $(i_1,\ldots,i_n)$ of integers in $\{1,\ldots,n\}$ as 0 if two of the integers are equal, and otherwise as the signature of the permutation formed by the tuple. Then

$\det(A)=\sum_{i_1,i_2,\ldots,i_n}\varepsilon_{i_1\cdots i_n}\,a_{1,i_1}\cdots a_{n,i_n}.$

Why the Leibniz formula satisfies the other definitions: one proves it is an alternating multilinear form taking the value 1 on the identity matrix.

• Multilinearity holds because the formula is a homogeneous polynomial of degree one in the entries of each column.
• Being alternating and equal to 1 on the identity follows from basic properties of permutations.

Alternating multilinear form

The determinant can also be defined as the unique alternating multilinear form of the columns (or rows) that takes the value 1 on the identity matrix.

Regard an $n\times n$ matrix as composed of its columns, $A=(a_1,\dots,a_n)$.

• Multilinear: if $A=(a_1,\dots,a_i,\ldots,a_n)$ and $A'=(a_1,\dots,a'_i,\ldots,a_n)$ differ in one column only, then for every scalar $r$

$\det(a_1,\dots,a_i+ra'_i,\ldots,a_n)=\det(a_1,\dots,a_i,\ldots,a_n)+r\det(a_1,\dots,a'_i,\ldots,a_n).$

• Alternating: the form takes the value 0 if two columns are equal.

Equivalence with the earlier definitions follows by checking the elementary-transformation properties, which is straightforward.

Over a commutative ring

The determinant is also defined for matrices with entries in a commutative ring $R$, and then it is an element of $R$. A common example is the Jacobian determinant, where entries and determinant are differentiable functions.

• The Leibniz formula and the alternating multilinear definition, and their equivalence, remain valid in this more general case. They imply the properties used in the multiplicative and elementary-transformation definitions.
• They also imply invariance under ring homomorphisms: if $f:R\to R'$ is a homomorphism of commutative rings and $f(M)$ is the matrix obtained by applying $f$ to the entries of $M$, then $\det(f(M))=f(\det(M))$.

Generic determinant

A generic matrix has distinct indeterminates as entries.

• By the Leibniz formula, the determinant of a generic matrix is a polynomial in these indeterminates with integer coefficients, called a generic determinant.
• For an $n\times n$ matrix with the $n^2$ indeterminates $X_{1,1},\cdots,X_{n,n}$, the entries and determinant belong to the polynomial ring $\mathbb{Z}[X_{1,1},\cdots,X_{n,n}]$ (and thus to its field of fractions).
• An $n\times n$ matrix $M$ over a commutative ring $R$ defines a ring homomorphism $\varphi_M:\mathbb{Z}[X_{1,1},\cdots,X_{n,n}]\to R$ mapping each $X_{i,j}$ to the corresponding entry of $M$. By invariance, $\varphi_M$ maps the generic determinant to $\det(M)$.
• Proof strategy: to prove properties over a commutative ring $R$ that is not an integral domain, it often suffices to prove them over fields, transfer them to integral domains by invariance under ring homomorphisms, and then transfer to $R$ using $\varphi_M$.
• The generic determinant is an irreducible polynomial.

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⚙️ Properties

Basic rules

These are parts or immediate consequences of the definitions and are often used implicitly.

Zero test. A determinant equals zero if:

• all entries of a row or column equal 0
• two rows or columns are equal
• a row or column is a linear combination of the other rows or columns
• a linear combination of rows or columns, with at least one nonzero coefficient, has all entries equal to zero

Immediate computation:

• The determinant of an identity matrix is 1.
• The determinant of a diagonal matrix is the product of its diagonal entries.
• The determinant of a triangular matrix is the product of its diagonal entries.

Simple transformations:

Operation	Effect on the determinant
Interchange two rows or two columns	Multiplies by $-1$
Multiply a row or column by a scalar (homogeneity)	Multiplies by the same scalar
Transpose the matrix	Unchanged
Add to a row or column a scalar multiple of another	Unchanged
Add to a row or column a linear combination of the others	Unchanged

Matrix multiplication:

• $\det(AB)=\det(A)\det(B)$.
• Similar matrices have the same determinant: if $B=S^{-1}AS$ for an invertible $S$, then $\det(B)=\det(A)$.

Invertibility:

• Over a field, a matrix is invertible if and only if its determinant is nonzero.
• Over a commutative ring, a matrix is invertible if and only if its determinant is invertible in the ring; the inverse's entries then also belong to the ring.
• A square matrix of integers has an integer inverse if and only if its determinant is $1$ or $-1$.

Laplace expansion

By the Leibniz formula, a determinant is a homogeneous polynomial of degree one in the elements of each row or column. For $A=(a_{i,j})$ of order $n$:

$\det A=\sum_{j=1}^{n}a_{i,j}A_{i,j},$

where $A_{i,j}$ is the cofactor of $a_{i,j}$, a polynomial in the other entries. This is the expansion along the $i$th row; expansion along the $j$th column is defined similarly.

The cofactors are given explicitly by

$A_{i,j}=(-1)^{i+j}M_{i,j},$

where $M_{i,j}$ is the minor: the determinant of the order $n-1$ matrix obtained by removing the $i$th row and $j$th column.

Example. Expansion along the first row of a $3\times 3$ matrix:

$\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}=a\begin{vmatrix}e&f\\h&i\end{vmatrix}-b\begin{vmatrix}d&f\\g&i\end{vmatrix}+c\begin{vmatrix}d&e\\g&h\end{vmatrix}$

Efficiency and use:

• Used iteratively, it is inefficient for large matrices. A naive implementation needs $n!$ sub-determinants; even memoizing them, there are $2^n-1$ sub-determinants, an exponential algorithm, whereas Gaussian elimination is polynomial time.
• It can be useful for proving by recursion formulas that depend on parameters.
• It is useful for sparse matrices. If $a_{i,j}$ is the unique nonzero entry of its row or column, then $\det A=(-1)^{i+j}M_{i,j}$.

Adjugate matrix

The adjugate $\operatorname{adj}(A)$ is the transpose of the matrix of cofactors:

$(\operatorname{adj}(A))_{i,j}=(-1)^{i+j}M_{j,i}.$

For every matrix,

$(\det A)I=A\operatorname{adj}A=(\operatorname{adj}A)\,A.$

So the inverse of a nonsingular matrix is

$A^{-1}=\frac{1}{\det A}\operatorname{adj}A.$

Invertibility in depth

• By the multiplicative property, over every commutative ring, if a matrix is invertible then its determinant is invertible.
• Conversely, by the adjugate formulas, if the determinant is invertible in a commutative ring containing the entries, the matrix is invertible in that ring.
• A matrix of integers is invertible over the integers iff its determinant is $\pm1$. Such a matrix is called unimodular.
• Invertible $n\times n$ matrices over a commutative ring $R$ form a group under multiplication, the general linear group $\operatorname{GL}_n(R)$. The most studied are $\operatorname{GL}_n(\mathbb{R})$, $\operatorname{GL}_n(\mathbb{C})$ and $\operatorname{GL}_n(\mathbb{Z})$.
• Their normal subgroups of determinant-1 matrices are the special linear groups $\operatorname{SL}_n(\mathbb{R})$, $\operatorname{SL}_n(\mathbb{C})$, $\operatorname{SL}_n(\mathbb{Z})$.

Invariance under ring homomorphisms

For a ring homomorphism $f:R\to S$ and a matrix $M$ over $R$:

$\det f(M)=f(\det M).$

• When $f$ is an inclusion, the determinant does not depend on whether the entries are viewed in $R$ or $S$. The determinant belongs to the smallest ring containing the entries.
• The determinant of an integer matrix is always an integer. For computing it one may use divisions, working over $\mathbb{Q}$: intermediate values may leave $\mathbb{Z}$, but the final result is in $\mathbb{Z}$ and is the same as without division.
• When $S=R$ and $f$ is an automorphism, the determinant of the conjugate of a complex matrix is the conjugate of its determinant.
• Modular arithmetic: reduce the entries modulo several primes $p_i$, compute the determinant in each field $\mathbb{Z}/p_i\mathbb{Z}$ (possibly with different methods), and recover the result with the Chinese remainder theorem.
• Category theory view: the determinant is a natural transformation between two functors from commutative rings to monoids. The first maps a ring to the monoid (under matrix multiplication) of order-$n$ square matrices over it; the second is the forgetful functor mapping a ring to its multiplicative monoid.

Hadamard bound

It is often useful to bound $\vert\det(A)\vert$ in terms of the size of the entries. Let $A=(a_{i,j})$ be $n\times n$ with complex entries whose absolute values are at most $B$.

• The triangle inequality applied to the Leibniz formula gives $\vert\det(A)\vert\leq n!B^{n}$.
• This is rarely used because the following bound is always sharper for $n>1$:

$\vert\det(A)\vert\leq n^{n/2}B^{n}.$

• Properly speaking, Hadamard inequality is

$\vert\det(A)\vert\leq\prod_{j=1}^{n}\|A_{j}\|,$

where $\|A_j\|=\sqrt{|a_{1,j}|^{2}+\cdots+|a_{n,j}|^{2}}$ is the quadratic norm of the $j$th column.

• Equality holds if and only if the columns are pairwise orthogonal, that is $\sum_{i=1}^{n}a_{i,j}a_{i,k}=0$ for $j\neq k$.
• Geometric counterpart: the volume of the parallelotope defined by $n$ vectors in an $n$-dimensional Euclidean space is at most the product of the vector lengths, with equality iff the vectors are pairwise orthogonal.
• If $A$ is positive definite, the determinant is bounded by the product of the main diagonal entries (no absolute value needed, since both are real and positive):

$\det(A)\leq\prod_{i=1}^{n}a_{i,i}.$

This is sometimes also called the Hadamard inequality.

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🧩 Other Determinants

Several mathematical objects can be represented by square matrices, which allows defining a determinant on them.

Object	Determinant
Endomorphism of a finite-dimensional vector space	Determinant of the matrix representing it on any basis
$n$ vectors in a space of dimension $n$	Determinant of the matrix of their coordinates on a given basis
Quadratic form over a field $F$	Discriminant: determinant of its matrix, defined up to multiplication by a square

Determinant of an endomorphism

• Two square matrices $A$ and $B$ represent the same endomorphism on different bases iff they are similar: $B=S^{-1}AS$.
• By the multiplicative property, $\det(B)=\det(A)$.
• So the determinant of an endomorphism is well defined and independent of the basis.

Determinant of n vectors

• Given a basis, the coordinates of each vector form a column; together they form a square matrix, whose determinant is the determinant of the $n$ vectors.
• It depends on the basis, so one specifies "on the given basis" when several bases are in play.
• In a Euclidean vector space one generally considers only orthonormal bases; then the determinant does not depend on the specific orthonormal basis.

Discriminant of a quadratic form

• A quadratic form over a field $F$ is a homogeneous polynomial of degree 2, naturally represented by a square matrix.
• The discriminant is that matrix's determinant, defined only up to a square, so it is an element of $F/(F^{\times})^{2}$, where $(F^{\times})^{2}$ is the set of nonzero squares in $F$.

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📈 Derivative of the Determinant

Let $A=(a_{i,j})$ be a square matrix of differentiable functions of $x$. Since $\det(A)$ is a polynomial in the entries, it is differentiable in $x$. Jacobi's formula:

$\frac{d}{dx}\det(A)=\operatorname{tr}\left(\operatorname{adj}(A)\frac{dA}{dx}\right),$

where $\operatorname{tr}$ is the trace (sum of diagonal entries), $\operatorname{adj}(A)$ the adjugate, and $dA/dx$ the matrix of derivatives of the entries.

• If $A$ is invertible:

$\frac{d}{dx}\det(A)=\det(A)\operatorname{tr}\left(A^{-1}\frac{dA}{dx}\right).$

• Partial derivative with respect to an entry $A_{i,j}$ (the last equality only for invertible $A$):

$\frac{\partial}{\partial A_{i,j}}\det(A)=\operatorname{adj}(A)_{j,i}=\det(A)\left(A^{-1}\right)_{j,i}.$

• Change under a small deformation:

$\det(A+\varepsilon X)-\det(A)=\operatorname{tr}(\operatorname{adj}(A)X)\varepsilon+O\left(\varepsilon^{2}\right)$

• For a deformation of the identity matrix:

$\det(I+\varepsilon X)=1+\operatorname{tr}(X)\varepsilon+O\left(\varepsilon^{2}\right).$

Lie algebra application. This identity is used to describe Lie algebras of certain matrix Lie groups. The special linear group $\operatorname{SL}_n$ is defined by $\det A=1$, which shows its Lie algebra is the special linear Lie algebra $\mathfrak{sl}_n$, consisting of the matrices with trace zero.

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🖥️ Computation

• Determinants are mainly a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for checking invertibility or finding eigenvalues other techniques have largely supplanted them.
• Computational geometry, however, frequently uses calculations related to determinants.
• The Leibniz rule is extremely inefficient for large matrices: it needs $n!$ products, so the number of operations grows on the order of $n!$. The Laplace expansion is similarly inefficient.

Gaussian elimination

• Left-multiply the matrix by elementary matrices to reach a row echelon form. Restricting to elementary matrices of determinant 1, the echelon form has the same determinant as the original.
• A row echelon form is triangular, so its determinant is the product of its diagonal entries.
• Hence the determinant comes almost for free from Gaussian elimination.

Decomposition methods

Write the matrix as a product of matrices with easily computed determinants. Examples: LU, QR, and Cholesky (for positive definite matrices). These are of order $O(n^3)$, a significant improvement over $O(n!)$.

LU example. Express $A=PLU$ with:

• $P$ a permutation matrix (exactly one 1 in each column, otherwise zeros)
• $L$ lower triangular
• $U$ upper triangular

The determinants of $L$ and $U$ are the products of their diagonal entries. $\det P$ is the sign $\varepsilon$ of the permutation ($+1$ for an even number of permutations, $-1$ for odd). Then

$\det(A)=\varepsilon\det(L)\cdot\det(U).$

Further methods

• Fast matrix multiplication. If two order-$n$ matrices can be multiplied in time $M(n)$, where $M(n)\geq n^{a}$ for some $a>2$, then the determinant can be computed in time $O(M(n))$. For example, an $O(n^{2.376})$ algorithm exists based on the Coppersmith–Winograd algorithm; the exponent was lowered, as of 2016, to 2.373.
• Division-free algorithms exist for matrices over rings (Gaussian elimination requires divisions). One has complexity $O(n^4)$ and replaces permutations (as in the Leibniz rule) by closed ordered walks, in which several items can be repeated. The sum has more terms, but many products can be reused.
• Bit complexity (bits needed to store intermediate values): Gaussian elimination / LU is $O(n^3)$, but the bit length of intermediate values can become exponentially long. The Bareiss algorithm, an exact-division method (divisions are performed only without remainder), has the same order but a bit complexity roughly the bit size of the original entries times $n$.
• Matrix determinant lemma: if $\det A$ and $A^{-1}$ are already known, it allows rapid calculation of the determinant of $A+uv^{T}$ for column vectors $u$ and $v$.
• Dodgson condensation, invented by Charles Dodgson (Lewis Carroll), computes determinants but does not always work in its original form.

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🧮 Uses in Linear Algebra

Determinants are a fundamental tool in linear algebra. They are rarely used in numerical computations, since they are typically a by-product of computations and no faster algorithm for direct computation is known. They are, however, unavoidable in proofs and when matrix entries are functions or depend on parameters.

• Cramer's rule: for a system of $n$ linearly independent equations in $n$ unknowns, each unknown is a quotient of two determinants formed with the coefficients.
• Adjugate matrix: its entries are determinants of submatrices. For an invertible matrix, each entry of the inverse is an adjugate entry divided by the determinant.
• Rank: the rank is at most $k$ if the determinant of every square submatrix of dimension larger than $k$ is zero. This is not a good way to compute rank, but it shows immediately that the matrices of rank at most $k$ among generic $m\times n$ matrices form an algebraic set (in fact an algebraic variety). This plays a fundamental role in the theories of Grassmannians and determinantal ideals.
• Hadamard's inequality: bounds the absolute values of solutions expressible with determinants. Starting from an integer matrix, it bounds the size of every integer in the solution. A common challenge is to design algorithms where no integer used exceeds this bound, as in the Bareiss algorithm and multi-modular arithmetic with Chinese remaindering.

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📏 Uses in Euclidean Geometry

Properties of points in Euclidean space expressed via determinants of their Cartesian coordinates.

Plane geometry

Let $P_0=(x_0,y_0)$, $P_1=(x_1,y_1)$, $P_2=(x_2,y_2)$ be three points.

• Colinear iff

$\det\begin{pmatrix}1&1&1\\x_{0}&x_{1}&x_{2}\\y_{0}&y_{1}&y_{2}\end{pmatrix}=0.$

• Subtracting the first column from the other two and expanding along the first row:

$\det\begin{pmatrix}1&1&1\\x_{0}&x_{1}&x_{2}\\y_{0}&y_{1}&y_{2}\end{pmatrix}=\det\begin{pmatrix}x_{1}-x_{0}&x_{2}-x_{0}\\y_{1}-y_{0}&y_{2}-y_{0}\end{pmatrix}.$

This is the determinant of the vectors $\overrightarrow{P_0P_1}$ and $\overrightarrow{P_0P_2}$.

• If the points are not aligned, the absolute value is twice the area of the triangle. The sign gives the triangle's orientation: positive iff $P_0$ is on the left when running from $P_1$ to $P_2$.
• Replacing $P_0$ with an indeterminate point $P=(X,Y)$ gives the implicit equation of the line through $P_1$ and $P_2$:

$\det\begin{pmatrix}1&1&1\\X&x_{1}&x_{2}\\Y&y_{1}&y_{2}\end{pmatrix}=0,$

which, expanding along the first column, becomes

$X(y_{2}-y_{1})+Y(x_{2}-x_{1})+(x_{1}y_{2}-x_{2}y_{1})=0.$

Dimension 3 and higher

Let $P_0,\dots,P_3$ be four points in three-dimensional space, $P_i=(x_i,y_i,z_i)$.

• Coplanar iff

$\det\begin{pmatrix}1&1&1&1\\x_{0}&x_{1}&x_{2}&x_{3}\\y_{0}&y_{1}&y_{2}&y_{3}\\z_{0}&z_{1}&z_{2}&z_{3}\end{pmatrix}=0.$

• Subtracting the first column from the others and expanding gives the $3\times3$ determinant of the vectors $\overrightarrow{P_0P_1}$, $\overrightarrow{P_0P_2}$, $\overrightarrow{P_0P_3}$.
• If not coplanar, the absolute value is six times the volume of the tetrahedron defined by the points. The sign gives the tetrahedron's orientation, and in particular the side of the plane $P_1P_2P_3$ on which $P_0$ lies. This criterion is commonly used in computer graphics and computer vision to recognize hidden parts of a 3D figure.
• Replacing $P_0$ with $P=(X,Y,Z)$ gives the implicit equation of the plane through three non-colinear points; in conventional form:

$X\begin{vmatrix}1&1&1\\y_{1}&y_{2}&y_{3}\\z_{1}&z_{2}&z_{3}\end{vmatrix}-Y\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\z_{1}&z_{2}&z_{3}\end{vmatrix}+Z\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\end{vmatrix}-\begin{vmatrix}x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\\z_{1}&z_{2}&z_{3}\end{vmatrix}=0.$

• In dimension $n$: $n+1$ points, "coplanar" becomes "in the same hyperplane", "tetrahedron" becomes "simplex", and the determinant is $n!$ times the volume of the simplex.

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🎯 Eigenvalues and Characteristic Polynomial

An eigenvalue of a linear endomorphism $f$ is a scalar $\lambda$ such that there is a nonzero vector $u$ (an eigenvector) with $f(u)=\lambda u$. For a matrix $A$: $AU=\lambda U$ for a nonzero column vector $U$.

• Equivalent: $(\lambda I-A)U=0$.
• This means the columns of $\lambda I-A$ are linearly dependent, i.e.

$\det(\lambda I-A)=0.$

• So the eigenvalues are the roots of $\det(XI-A)$, the characteristic polynomial of $A$.
• Being a determinant, it is invariant under similarity, hence also defined for linear endomorphisms.

For $A$ of order $n$:

• The characteristic polynomial is monic of degree $n$, and all its coefficients are similarity invariants.
• Its constant coefficient (set $X=0$) is $(-1)^{n}\det(A)$.
• The coefficient of $X^{n-1}$ is $-\operatorname{Tr}(A)$. By Viete's formula it is the sum of the eigenvalues (with multiplicities). By the Leibniz formula the trace is also the sum of the diagonal entries of $A$ and of every matrix similar to $A$.

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🔬 Determinants of Functions

Jacobian determinant

The Jacobian matrix of differentiable functions $f_1,\ldots,f_n$ in $m$ variables $x_1,\ldots,x_m$ is the matrix of partial derivatives:

$\begin{pmatrix}\tfrac{\partial f_{1}}{\partial x_{1}}&\ldots&\tfrac{\partial f_{n}}{\partial x_{1}}\\\vdots&\cdots&\vdots\\\tfrac{\partial f_{1}}{\partial x_{m}}&\ldots&\tfrac{\partial f_{n}}{\partial x_{m}}\end{pmatrix}.$

• When $m=n$, its determinant is the Jacobian determinant, a function of the $n$ variables.
• The Jacobian matrix generalizes the derivative to multivariate vector-valued functions; the condition $f'(x)\neq0$ is replaced by $J\neq0$.
• Two especially important uses:
  • it occurs in the formula for change of variables in a multiple integral
  • its non-nullity is a hypothesis of the implicit function theorem and the inverse function theorem

Wronskian

For $n-1$ times differentiable functions $f_1,\dots,f_n$:

$W(f_1,\ldots,f_n)(x)=\begin{vmatrix}f_1(x)&f_2(x)&\cdots&f_n(x)\\f_1'(x)&f_2'(x)&\cdots&f_n'(x)\\\vdots&\vdots&\ddots&\vdots\\f_1^{(n-1)}(x)&f_2^{(n-1)}(x)&\cdots&f_n^{(n-1)}(x)\end{vmatrix}.$

If the Wronskian is zero everywhere on an interval then, for analytic functions, the functions are linearly dependent.

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🧭 Uses in Vector Geometry

In a Euclidean vector space (a real vector space, typically $\mathbb{R}^n$ with the usual quadratic norm).

Orientation

• The determinant of $n$ vectors in $\mathbb{R}^n$ is nonzero iff they form a basis. Its sign tells whether the orientation of the basis is the same as, or opposite to, the standard basis.
• For an orthogonal basis, the absolute value equals the product of the basis vector lengths.
• For an orthonormal basis, the determinant is $1$ (same orientation as the standard basis) or $-1$ (otherwise).
• A linear transformation preserves orientation if its determinant is positive and reverses it if negative. This does not depend on the choice of basis.

Signed volume of a parallelotope

The parallelotope generated by $v_1,\ldots,v_n$ is the set of vectors $a_1v_1+\cdots+a_nv_n$ with $0\leq a_i\leq1$.

• Given an orthonormal basis, its $n$-dimensional volume is the absolute value of the determinant of the vectors on that basis. The volume is zero if the vectors are linearly dependent.
• For independent vectors, the determinant is positive if their orientation matches the basis, negative otherwise.
• The parallelotope generated by an orthonormal basis is a unit hypercube with volume 1.

For a bijective linear transformation $f$ with determinant $D$:

• $D>0$: $f$ preserves space orientation
• $D<0$: $f$ reverses space orientation
• $f$ multiplies every $n$-dimensional volume by $\vert D\vert$

Volume and Jacobian determinant

If $f:\mathbb{R}^n\to\mathbb{R}^n$ is multiplication by a matrix $A$ and $S$ is a measurable subset, then the volume of $f(S)$ is $|\det(A)|$ times the volume of $S$.

More generally, if $f:\mathbb{R}^n\to\mathbb{R}^m$ is represented by an $m\times n$ matrix $A$, the ratio of the $n$-dimensional volumes of $f(S)$ and $S$ is

$\frac{\operatorname{volume}(f(S))}{\operatorname{volume}(S)}=\sqrt{\det\left(A^{\mathsf T}A\right)}.$

When $m<n$ this is zero.

• By computing the volume of the tetrahedron bounded by four points, determinants can identify skew lines. The volume of a tetrahedron with vertices $a,b,c,d$ is $\frac{1}{6}\cdot|\det(a-b,b-c,c-d)|$, or any other combination of pairs of vertices forming a spanning tree over the vertices.
• For a general differentiable $f:\mathbb{R}^n\to\mathbb{R}^n$, the Jacobian matrix is $D(f)=\left(\frac{\partial f_i}{\partial x_j}\right)_{1\leq i,j\leq n}$. Its determinant appears in the higher-dimensional integration by substitution: for suitable $f$ and an open subset $U$ of $\mathbb{R}^n$,

$\int_{f(U)}\phi(\mathbf{v})\,d\mathbf{v}=\int_{U}\phi(f(\mathbf{u}))\left|\det(\operatorname{D}f)(\mathbf{u})\right|\,d\mathbf{u}.$

• The Jacobian also occurs in the inverse function theorem.
• In cartography, the determinant can measure the rate of expansion of a map near the poles.

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🌐 Generalizations

• The immanant replaces $\operatorname{sgn}(\sigma)$ in the Leibniz formula with another character of the symmetric group $S_n$. A special case is the permanent, which replaces $\operatorname{sgn}(\sigma)$ by the constant 1.

Exterior algebra

The determinant of a linear transformation $T:V\to V$ of an $n$-dimensional vector space (or a free module of rank $n$ over a commutative ring $R$) can be formulated without coordinates using the $n$-th exterior power $\bigwedge^{n}V$.

• $T$ induces a linear map $\bigwedge^{n}T$ sending $v_1\wedge\dots\wedge v_n$ to $Tv_1\wedge\dots\wedge Tv_n$.
• Since $\bigwedge^{n}V$ is one-dimensional, this map is multiplication by a scalar in $R$. Some authors (e.g. Bourbaki) define the determinant as this scalar:

$\left(\bigwedge^{n}T\right)\left(v_{1}\wedge\dots\wedge v_{n}\right)=\det(T)\cdot v_{1}\wedge\dots\wedge v_{n}.$

• This agrees with the coordinate definition, using the uniqueness of the multilinear alternating form on $n$-tuples of vectors in $R^n$.
• The highest nonzero exterior power $\bigwedge^{n}V$ is sometimes itself called the determinant of $V$, and similarly for objects such as vector bundles or chain complexes. Minors can be treated via lower alternating forms $\bigwedge^{k}V$, $k<n$.

Finite-dimensional algebras

For any associative algebra $A$ that is finite-dimensional as a vector space over a field $F$, there is a determinant map

$\det:A\to F.$