In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. A scale factor is a number which scales, or multiplies some quantity The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Mathematics, multilinear algebra extends the methods of Linear algebra.

For a fixed positive integer n, there is a unique determinant function for the n×n matrices over any commutative ring R. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In particular, this function exists when R is the field of real or complex numbers. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Contents 1 Vertical bar notation 2 Determinants of 2-by-2 matrices 3 Determinants of 3-by-3 matrices 4 Applications 5 General definition and computation 6 Example 7 Properties 7.1 Useful identities 7.2 Block matrices 7.3 Relationship to trace 7.4 Derivative 8 Abstract formulation 9 Algorithmic implementation 10 History 11 See also 12 References 13 External links

Vertical bar notation

The determinant of a matrix A is also sometimes denoted by |A|. This notation can be ambiguous since it is also used for certain matrix norms and for the absolute value. In Mathematics, a matrix norm is a natural extension of the notion of a Vector norm to matrices. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. However, often the matrix norm will be denoted with double vertical bars (e. g. , ||A||) and may carry a subscript as well. Thus, the vertical bar notation for determinant is frequently used (e. g. , Cramer's rule and minors). Cramer's rule is a Theorem in Linear algebra, which gives the solution of a System of linear equations in terms of Determinants It is named after In Linear algebra, a minor of a matrix A is the Determinant of some smaller Square matrix, cut down from A by removing For example, for matrix

the determinant det(A) might be indicated by | A | or more explicitly as

That is, the square braces around the matrices are replaced with elongated vertical bars.

Determinants of 2-by-2 matrices

The 2×2 matrix,

has determinant

The interpretation when the matrix has real number entries is that this gives the oriented area of the parallelogram with vertices at (0,0), (a,b), (a + c, b + d), and (c,d). In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides The oriented area is the same as the usual area, except that it is negative when the vertices are listed in clockwise order.

The assumption here is that a linear transformation is applied to row vectors as the vector-matrix product x^TA, where x is a column vector. The parallelogram in the figure is obtained by multiplying matrix A (which stores the co-ordinates of our parallelogram) with each of the row vectors and in turn. These row vectors define the vertices of the unit square. With the more common matrix-vector product Ax, the parallelogram has vertices at and (note that Ax = (x^TA^T)^T).

A formula for larger matrices will be given below.

Determinants of 3-by-3 matrices

The 3×3 matrix:

Using the cofactor expansion on the first row of the matrix we get:

which can be remembered as the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements when the copies of the first two columns of the matrix are written beside it as below:

Note that this mnemonic does not carry over into higher dimensions.

Applications

Determinants are used to characterize invertible matrices (i. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- e. , exactly those matrices with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable Cramer's rule is a Theorem in Linear algebra, which gives the solution of a System of linear equations in terms of Determinants It is named after They can be used to find the eigenvalues of the matrix A through the characteristic polynomial

where I is the identity matrix of the same dimension as A. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Linear algebra, one associates a Polynomial to every Square matrix, its characteristic polynomial. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main

One often thinks of the determinant as assigning a number to every sequence of n vectors in , by using the square matrix whose columns are the given vectors. In Mathematics, a sequence is an ordered list of objects (or events With this understanding, the sign of the determinant of a basis can be used to define the notion of orientation in Euclidean spaces. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which The determinant of a set of vectors is positive if the vectors form a right-handed coordinate system, and negative if left-handed. A negative number is a Number that is less than zero, such as −2 In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point

Determinants are used to calculate volumes in vector calculus: the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism As a consequence, if the linear map is represented by the matrix A, and S is any measurable subset of , then the volume of f(S) is given by . In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to More generally, if the linear map is represented by the m-by-n matrix A, and S is any measurable subset of , then the m-dimensional volume of f(S) is given by . In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. In Geometry, skew lines are two lines that do not intersect but are not Parallel.

The volume of any tetrahedron, given its vertices a, b, c, and d, is (1/6)·|det(a − b, b − c, c − d)|, or any other combination of pairs of vertices that form a simply connected graph. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects

General definition and computation

The definition of the determinant comes from the following Theorem.

Theorem. Let M_n(K) denote the set of all matrices over the field K. There exists exactly one function

with the two properties:

• F is alternating multilinear with regard to columns;
• F(I) = 1. In Linear algebra, a multilinear map is a Mathematical function of several vector variables that is linear in each variable

One can then define the determinant as the unique function with the above properties.

In proving the above theorem, one also obtains the Leibniz formula:

Here the sum is computed over all permutations σ of the numbers {1,2,. In Algebra, the Leibniz formula expresses the Determinant of a square matrix A = (a_{ij}_{ij = 1 \dots n} in terms of permutations of the matrix' In several fields of Mathematics the term permutation is used with different but closely related meanings . . ,n} and sgn(σ) denotes the signature of the permutation σ: +1 if σ is an even permutation and −1 if it is odd. In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even σ: can also denote the signature of the number of inversions of the product of the permutation which is the approach used in some textbooks.

This formula contains n! (factorial) summands, and it is therefore impractical to use it to calculate determinants for large n. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k

For small matrices, one obtains the following formulas:

• if A is a 1-by-1 matrix, then

• if A is a 2-by-2 matrix, then

• for a 3-by-3 matrix A, the formula is more complicated:

which takes the shape of the Sarrus' scheme. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

In general, determinants can be computed using Gaussian elimination using the following rules:

• If A is a triangular matrix, i. In Linear algebra, Gaussian elimination is an efficient Algorithm for solving systems of linear equations, to find the rank of a matrix In the mathematical discipline of Linear algebra, a triangular matrix is a special kind of Square matrix where the entries either below or above the e. whenever i > j or, alternatively, whenever i < j, then (the product of the diagonal entries of A).
• If B results from A by exchanging two rows or columns, then
• If B results from A by multiplying one row or column with the number c, then
• If B results from A by adding a multiple of one row to another row, or a multiple of one column to another column, then

Explicitly, starting out with some matrix, use the last three rules to convert it into a triangular matrix, then use the first rule to compute its determinant.

It is also possible to expand a determinant along a row or column using Laplace's formula, which is efficient for relatively small matrices. In Linear algebra, the Laplace expansion of the Determinant ofan n  ×  n square matrix B expresses the determinant To do this along row i, say, we write

where the C_i,j represent the matrix cofactors, i. In Linear algebra, a minor of a matrix A is the Determinant of some smaller Square matrix, cut down from A by removing e. C_i,j is ( − 1)^{i + j} times the minor M_i,j, which is the determinant of the matrix that results from A by removing the i-th row and the j-th column. In Linear algebra, a minor of a matrix A is the Determinant of some smaller Square matrix, cut down from A by removing

Example

Suppose we want to compute the determinant of

We can go ahead and use the Leibniz formula directly:

Alternatively, we can use Laplace's formula to expand the determinant along a row or column. In Linear algebra, the Laplace expansion of the Determinant ofan n  ×  n square matrix B expresses the determinant It is best to choose a row or column with many zeros, so we will expand along the second column:

A third way (and the method of choice for larger matrices) would involve the Gauss algorithm. When doing computations by hand, one can often shorten things dramatically by cleverly adding multiples of columns or rows to other columns or rows; this does not change the value of the determinant, but may create zero entries which simplifies the subsequent calculations. In this example, adding the second column to the first one is especially useful:

and this determinant can be quickly expanded along the first column:

Properties

The determinant is a multiplicative map in the sense that

for all n-by-n matrices A and B.

This is generalized by the Cauchy-Binet formula to products of non-square matrices. In Linear algebra, the Cauchy-Binet formula generalizes the multiplicativity of the Determinant (the fact that the determinant of a product of two

It is easy to see that and thus

for all n-by-n matrices A and all scalars r. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication

A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In particular, if A is a matrix over a field such as the real or complex numbers, then A is invertible if and only if det(A) is not zero. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In this case we have

Expressed differently: the vectors v₁,. . . ,v_n in Rⁿ form a basis if and only if det(v₁,. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. . . ,v_n) is non-zero.

A matrix and its transpose have the same determinant:

The determinants of a complex matrix and of its conjugate transpose are conjugate:

(Note the conjugate transpose is identical to the transpose for a real matrix)

The determinant of a matrix A exhibits the following properties under elementary matrix transformations of A:

1. Exchanging rows or columns multiplies the determinant by −1. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a In Mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m -by- n matrix A with In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. In Mathematics, an elementary matrix is a simple matrix which differs from the Identity matrix in a minimal way
2. Multiplying a row or column by m multiplies the determinant by m.
3. Adding a multiple of a row or column to another leaves the determinant unchanged.

This follows from the multiplicative property and the determinants of the elementary matrix transformation matrices. In Mathematics, an elementary matrix is a simple matrix which differs from the Identity matrix in a minimal way

If A and B are similar, i. In Linear algebra, two n -by- n matrices A and B over the field K are called similar if there exists e. , if there exists an invertible matrix X such that A = X ^{− 1}BX, then by the multiplicative property,

This means that the determinant is a similarity invariant. In Mathematics, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain Because of this, the determinant of some linear transformation T : V → V for some finite dimensional vector space V is independent of the basis for V. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The relationship is one-way, however: there exist matrices which have the same determinant but are not similar.

If A is a square n-by-n matrix with real or complex entries and if λ₁,. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted . . ,λ_n are the (complex) eigenvalues of A listed according to their algebraic multiplicities, then

This follows from the fact that A is always similar to its Jordan normal form, an upper triangular matrix with the eigenvalues on the main diagonal. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Linear algebra, Jordan normal form (often called Jordan canonical form)shows that a given square matrix M over a field K

Useful identities

Sylvester's determinant theorem states that for any m-by-n matrices A and B,

For the case of (column) vectors a and b, this equality becomes

With X a nonsingular m-by-m matrix, this last expression generalizes to

Proofs can be found in [1]. In matrix theory Sylvester's determinant theorem is a theorem useful for evaluating certain types of Determinants It is named after James Joseph Sylvester

Block matrices

Suppose, A,B,C,D are matrices respectively. Then

This can be (quite) easily seen from e. g. the Leibniz formula. In Algebra, the Leibniz formula expresses the Determinant of a square matrix A = (a_{ij}_{ij = 1 \dots n} in terms of permutations of the matrix' Employing the following identity

leads to

Similar identity with det(D) factored out can be derived analogously. These identities were taken from [2].

If d_ij are diagonal matrices, then

This is a special case of the theorem published in [3].

Relationship to trace

From this connection between the determinant and the eigenvalues, one can derive a connection between the trace function, the exponential function, and the determinant:

Performing the substitution in the above equation yields

which is closely related to the Fredholm determinant. In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal In Mathematics, the matrix exponential is a Matrix function on square matrices analogous to the ordinary Exponential function. In Mathematics, the Fredholm determinant is a complex-valued function which generalizes the Determinant of a matrix. Similarly,

For n-by-n matrices there are the relationships:

Case n = 1:

Case n = 2:

Case n = 3:

Case n = 4:

which are closely related to Newton's identities. In Mathematics, Newton's identities, also known as the Newton–Girard formulae give relations between two types of Symmetric polynomials namely between power

Derivative

The determinant of real square matrices is a polynomial function from to , and as such is everywhere differentiable. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change Its derivative can be expressed using Jacobi's formula:

where adj(A) denotes the adjugate of A. In Matrix calculus, Jacobi's formula expresses the differential of the Determinant of a matrix A in terms of the Adjugate of A and In Linear algebra, the adjugate or classical adjoint of a Square matrix is a matrix which plays a role similar to the inverse of a matrix; it In particular, if A is invertible, we have

In component form, these are

When ε is a small number these are equivalent to

The special case where A is equal to the identity matrix I yields

A useful property in the case of 3 x 3 matrices is the following:

A may be written as where , , are vectors, then the gradient over one of the three vectors may be written as the cross product of the other two:

Abstract formulation

An n × n square matrix A may be thought of as the coordinate representation of a linear transformation of an n-dimensional vector space V. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Given any linear transformation

we can define the determinant of A as the determinant of any matrix representation of A. This is a well-defined notion (i. In Mathematics, the term well-defined is used to specify that a certain concept or object (a function, a property, a relation, etc e. independent of a choice of basis) since the determinant is invariant under similarity transformations. Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

As one might expect, it is possible to define the determinant of a linear transformation in a coordinate-free manner. If V is an n-dimensional vector space, then one can construct its top exterior power ΛⁿV. This is a one-dimensional vector space whose elements are written

where each v_i is a vector in V and the wedge product ∧ is antisymmetric (i. e. , u∧u = 0). Any linear transformation A : V → V induces a linear transformation of ΛⁿV as follows:

Since ΛⁿV is one-dimensional this operation is just multiplication by some scalar that depends on A. This scalar is called the determinant of A. That is, we define det(A) by the equation

One can check that this definition agrees with the coordinate-dependent definition given above.

Algorithmic implementation

• The naive method of implementing an algorithm to compute the determinant is to use Laplace's formula for expansion by cofactors. This approach is extremely inefficient in general, however, as it is of order n! (n factorial) for an n×n matrix M. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments Definition The factorial function is formally defined by n!=\prod_{k=1}^n k
• An improvement to order n³ can be achieved by using LU decomposition to write M = LU for triangular matrices L and U. In Linear algebra, the LU decomposition is a Matrix decomposition which writes a matrix as the product of a lower and upper Triangular matrix Now, det M = det LU = det L det U, and since L and U are triangular the determinant of each is simply the product of its diagonal elements. Alternatively one can perform the Cholesky decomposition if possible or the QR decomposition and find the determinant in a similar fashion. In Mathematics, the Cholesky decomposition is named after André-Louis Cholesky, who found that a symmetric Positive-definite matrix can be In Linear algebra, the QR decomposition (also called the QR factorization) of a matrix is a decomposition of the matrix into an orthogonal
• Since the definition of the determinant does not need divisions, a question arises: do fast algorithms exist that do not need divisions? This is especially interesting for matrices over rings. Indeed algorithms with run-time proportional to n⁴ exist. An algorithm of Mahajan and Vinay, and Berkowitz is based on closed ordered walks (short clow). It computes more products than the determinant definition requires, but some of these products cancel and the sum of these products can be computed more efficiently. The final algorithm looks very much like an iterated product of triangular matrices.
• What is not often discussed is the so-called "bit complexity" of the problem, i. e. how many bits of accuracy you need to store for intermediate values. For example, using Gaussian elimination, you can reduce the matrix to upper triangular form, then multiply the main diagonal to get the determinant (this is essentially a special case of the LU decomposition as above), but a quick calculation will show that the bit size of intermediate values could potentially become exponential. In Linear algebra, Gaussian elimination is an efficient Algorithm for solving systems of linear equations, to find the rank of a matrix One could talk about when it is appropriate to round intermediate values, but an elegant way of calculating the determinant uses the Bareiss Algorithm, an exact-division method based on Sylvester's identity to give a run time of order n³ and bit complexity roughly the bit size of the original entries in the matrix times n. In matrix theory Sylvester's determinant theorem is a theorem useful for evaluating certain types of Determinants It is named after James Joseph Sylvester

History

Historically, determinants were considered before matrices. Originally, a determinant was defined as a property of a system of linear equations. In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, determinants were first used in the 3rd century BC Chinese math textbook The Nine Chapters on the Mathematical Art. The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and In Europe, two-by-two determinants were considered by Cardano at the end of the 16th century and larger ones by Leibniz and, in Japan, by Seki about 100 years later. or (born 1637/1642? – October 24, 1708) was a Japanese Mathematician who created a new algebraic notation system and laid ^[1][2]

In Japan, determinants were introduced to study elimination of variables in systems of higher-order algebraic equations. In Commutative algebra and Algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between Polynomials of They used it to give short-hand representation for the resultant. In Mathematics, the resultant of two Monic polynomials P and Q over a field k is defined as the product After the first work by Seki in 1683, Laplace's formula was given by two independent groups of scholars: Tanaka, Iseki (算法発揮,Sampo-Hakki, published in 1690) and Seki, Takebe, Takebe (大成算経, taisei-sankei, written at least before 1710). or (born 1637/1642? – October 24, 1708) was a Japanese Mathematician who created a new algebraic notation system and laid In Linear algebra, the Laplace expansion of the Determinant ofan n  ×  n square matrix B expresses the determinant or (born 1637/1642? – October 24, 1708) was a Japanese Mathematician who created a new algebraic notation system and laid However, doubts have been raised about how much they recognized the determinant as an independent object.

In Europe, Cramer (1750) added to the theory, treating the subject in relation to sets of equations. Gabriel Cramer ( July 31, 1704 - January 4, 1752) was a Swiss Mathematician, born in Geneva. The recurrent law was first announced by Bézout (1764). Étienne Bézout ( March 31, 1730 - September 27, 1783) was a French Mathematician who was born in Nemours,

It was Vandermonde (1771) who first recognized determinants as independent functions. Alexandre-Théophile Vandermonde ( 28 February 1735 – 1 January 1796) was a French Musician and Chemist who ^[1] Laplace (1772) ^[3][4] gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case. In Linear algebra, a minor of a matrix A is the Determinant of some smaller Square matrix, cut down from A by removing Immediately following, Lagrange (1773) treated determinants of the second and third order. Lagrange was the first to apply determinants to questions of elimination theory; he proved many special cases of general identities. In Commutative algebra and Algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between Polynomials of

Gauss (1801) made the next advance. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Like Lagrange, he made much use of determinants in the theory of numbers. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes He introduced the word determinants (Laplace had used resultant), though not in the present signification, but rather as applied to the discriminant of a quantic. In Algebra, the discriminant of a Polynomial with real or complex Coefficients is a certain expression in the coefficients of the In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.

The next contributor of importance is Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of m columns and n rows, which for the special case of m = n reduces to the multiplication theorem. Jacques Philippe Marie Binet ( February 2, 1786 - May 12, 1856) was a French mathematician physicist and astronomer born in Rennes On the same day (November 30, 1812) that Binet presented his paper to the Academy, Cauchy also presented one on the subject. Events 1700 - Battle of Narva — A Swedish army of 8500 men under Charles XII defeats Year 1812 ( MDCCCXII) a leap year started on Wednesday (link will display the full calendar of the Gregorian calendar (or a Leap year (See Cauchy-Binet formula. In Linear algebra, the Cauchy-Binet formula generalizes the multiplicativity of the Determinant (the fact that the determinant of a product of two ) In this he used the word determinant in its present sense^[5][6], summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's. ^[1][7] With him begins the theory in its generality.

The next important figure was Jacobi^[2] (from 1827). Carl Gustav Jacob Jacobi ( December 10, 1804 - February 18, 1851) was a Prussian Mathematician, widely considered to be He early used the functional determinant which Sylvester later called the Jacobian, and in his memoirs in Crelle for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called alternants. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. Crelle's Journal, or just Crelle, is the common name for a leading German -language Mathematical journal, the Journal für About the time of Jacobi's last memoirs, Sylvester (1839) and Cayley began their work. James Joseph Sylvester ( September 3, 1814 London – March 15, 1897 Oxford) was an English Mathematician Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. ^[8][9]

The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessians by Sylvester; and symmetric gauche determinants by Trudi. Henri Léon Lebesgue leɔ̃ ləˈbɛg ( June 28, 1875, Beauvais &ndash July 26, 1941, Paris) was a French Ludwig Otto Hesse ( 22 April 1811 &ndash 4 August 1874) was a German Mathematician. In Mathematics, persymmetric matrix may refer to a Square matrix which is symmetric in the northeast-to-southwest diagonal or a square matrix such Hermann Hankel ( February 14, 1839 - August 29, 1873) was a German Mathematician who was born in Halle, In Linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each Row vector is rotated one element to the right relative to the preceding Eugène Charles Catalan ( May 30, 1814 &ndash February 14, 1894) was a Belgian Mathematician. William Spottiswoode FRS ( January 11 1825, London - June 27 1883 London) was an English mathematician and James Whitbread Lee Glaisher ( 5 November 1848 - 7 December 1928) son of James Glaisher, the meteorologist was a prolific English In Mathematics, the Determinant of a Skew-symmetric matrix can always be written as the square of a Polynomial in the matrix entries In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, the Wronskian is a function named after the Polish mathematician Józef Hoene-Wroński. Sir Thomas Muir ( 25 August 1844 – 21 March 1934) was a Scottish Mathematician, remembered as an authority on Determinants Elwin Bruno Christoffel ( November 10, 1829 in Montjoie now called Monschau – March 15, 1900 in Strasbourg) was a Ferdinand Georg Frobenius ( October 26, 1849 – August 3, 1917) was a German Mathematician, best-known for his contributions Reiss is a village in the former county of Caithness, now in the Highland Region of Northern Scotland. The picquet (alternately spelled piquet) was a military punishment in vogue in late Medieval Europe that was sufficiently cruel and ingenious to be characterized In Mathematics, the Hessian matrix is the Square matrix of second-order Partial derivatives of a function. Of the text-books on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.

See also

• Sylvester's determinant theorem
• Matrix determinant lemma
• Permanent
• Minor (linear algebra)

References

External links

• MIT Linear Algebra Lecture on Determinants
• Linear Systems Chapter from "Fundamental Problems of Algorithmic Algebra" Chee Yap's chapter on Linear Systems describing implementation aspects of Determinant computation.
• Mahajan, Meena and V. Vinay, “Determinant: Combinatorics, Algorithms, and Complexity”, Chicago Journal of Theoretical Computer Science, v. 1997 article 5 (1997).
• Online Matrix Calculator Online Matrix calculator.
• Linear algebra: determinants. Compute determinants of matrices up to order 6 using Laplace expansion you choose.

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