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By the name of Allah

The Determinant

Edit by:

Ahmed osman

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Determinant:

In linear algebra, the determinant is a valueassociated with a square matrix. It can be computedfrom the entries of the matrix by a specific arithmeticexpression, while other ways to determine its valueexist as well. The determinant provides importantinformation when the matrix is that of the coefficientsof a system of linear equations, or when itcorresponds to a linear transformation of a vectorspace: in the first case the system has a uniquesolution if and only if the determinant is nonzero, inthe second case that same condition means that thetransformation has an inverse operation. Ageometric interpretation can be given to the value ofthe determinant of a square matrix with real entries:

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the absolute value of the determinant gives the scalefactor by which area or volume is multiplied underthe associated linear transformation, while its signindicates whether the transformation preservesorientation. Thus a 2 2 matrix with determinant 2,when applied to a region of the plane with finitearea, will transform that region into one with twicethe area, while reversing its orientation.Determinants occur throughout mathematics. Theuse of determinants in calculus includes theJacobian determinant in the substitution rule forintegrals of functions of several variables. They areused to define the characteristic polynomial of amatrix that is an essential tool in eigenvalueproblems in linear algebra. In some cases they areused just as a compact notation for expressions thatwould otherwise be unwieldy to write down.

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The determinant of a matrix A is denoted det(A), detA, or |A|.[1] In the case where the matrix entries arewritten out in full, the determinant is denoted bysurrounding the matrix entries by vertical barsinstead of the brackets or parentheses of the matrix.For instance, the determinant of the matrix

is written and has the value.

Although most often used for matrices whose entriesare real or complex numbers, the definition of thedeterminant only involves addition, subtraction andmultiplication, and so it can be defined for squarematrices with entries taken from any commutativering. Thus for instance the determinant of a matrixwith integer coefficients will be an integer, and thematrix has an inverse with integer coefficients if andonly if this determinant is 1 or 1 (these being the

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onlyinvertibleelements oftheintegers).

For square matrices with entries in a non-commutative ring, for instance the quaternions, thereis no unique definition for the determinant, and nodefinition that has all the usual properties ofdeterminants over commutative rings.

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1 Definitiono 1.1 2-by-2 matriceso 1.2 3-by-3 matriceso 1.3 n-by-n matrices

1.3.1 Levi-Civita symbol 2 Properties of the determinant

o 2.1 Multiplicativity and matrix groupso 2.2 Laplace's formula and the adjugate

matrixo 2.3 Sylvester's determinant theorem

3 Properties of the determinant in relation toother notions

o 3.1 Relation to eigenvalues and traceo 3.2 Cramer's ruleo 3.3 Block matriceso 3.4 Derivative

4 Abstract algebraic aspectso 4.1 Determinant of an endomorphism

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o 4.2 Exterior algebrao 4.3 Square matrices over commutative rings

and abstract properties 5 Generalizations and related notions

o 5.1 Infinite matriceso 5.2 Notions of determinant over non-

commutative ringso 5.3 Further variants

6 Calculationo 6.1 Decomposition methodso 6.2 Further methods

7 History 8 Applications

o 8.1 Linear independenceo 8.2 Orientation of a basiso 8.3 Volume and Jacobian determinanto 8.4 Vander monde determinant (alternant)o 8.5 Circulates

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]Definition

There are various ways to define the determinant ofa square matrix A, i.e. one with the same number ofrows and columns. Perhaps the most natural way isexpressed in terms of the columns of the matrix. Ifwe write an n-by-n matrix in terms of its columnvectors

where the are vectors of size n, then thedeterminant of A is defined so that

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where b and c are scalars, v is any vector of size nand I is the identity matrix of size n. Theseproperties state that the determinant is analternating multilinear function of the columns, andthey suffice to uniquely calculate the determinant ofany square matrix. Provided the underlying scalarsform a field (more generally, a commutative ring withunity), the definition below shows that such afunction exists, and it can be shown to be unique. [2]Equivalently, the determinant can be expressed as asum of products of entries of the matrix where eachproduct has n terms and the coefficient of eachproduct is -1 or 1 or 0 according to a given rule: it isa polynomial expression of the matrix entries. Thisexpression grows rapidly with the size of the matrix(an n-by-n matrix contributes n! terms), so it will first

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be given explicitly for the case of 2-by-2 matricesand 3-by-3 matrices, followed by the rule forarbitrary size matrices, which subsumes these twocases.Assume A is a square matrix with n rows and ncolumns, so that it can be written as

The entries can be numbers or expressions (ashappens when the determinant is used to define acharacteristic polynomial); the definition of thedeterminant depends only on the fact that they canbe added and multiplied together in a commutativemanner.

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The determinant of A is denoted as det(A), or it canbe denoted directly in terms of the matrix entries bywriting enclosing bars instead of brackets:

 2-by-2 matrices

The area of the parallelogram is the absolute valueof the determinant of the matrix formed by thevectors representing the parallelogram's sides.The determinant of a 22 matrix is defined by

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If the matrix entries are real numbers, the matrix Acan be used to represent two linear mappings: onethat maps the standard basis vectors to the rows ofA, and one that maps them to the columns of A. Ineither case, the images of the basis vectors form aparallelogram that represents the image of the unitsquare under the mapping. The parallelogramdefined by the rows of the above matrix is the onewith vertices at (0,0), (a,b), (a + c, b + d), and (c,d),as shown in the accompanying diagram. Theabsolute value of is the area of theparallelogram, and thus represents the scale factorby which areas are transformed by A. (Theparallelogram formed by the columns of A is ingeneral a different parallelogram, but since the

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incompatible for n 2,[3] so there is no gooddefinition of the determinant in this setting.

4.A matrix and its transpose have the samedeterminant. This implies that properties forcolumns have their counterparts in terms ofrows:

5.Viewing an nn matrix as being composed of nrows, the determinant is an n-linear function.

6.This n-linear function is an alternating form:whenever two rows of a matrix are identical, itsdeterminant is 0.

7.Interchanging two columns of a matrix multipliesits determinant by 1. This follows fromproperties 2 and 3 (it is a general property ofmultilinear alternating maps). Iterating gives thatmore generally a permutation of the columnsmultiplies the determinant by the sign of thepermutation. Similarly a permutation of the rows

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Here, B is obtained from A by adding 1/2 the firstrow to the second, so that det(A) = det(B). C isobtained from B by adding the first to the third row,so that det(C) = det(B). Finally, D is obtained from Cby exchanging the second and third row, so thatdet(D) = det(C). The determinant of the (upper)triangular matrix D is the product of its entries on themain diagonal: (2) 2 4.5 = 18. Therefore det(A)= det(D) = +18. Multiplicativity and matrix groupsThe determinant of a matrix product of squarematrices equals the product of their determinants:

Thus the determinant is a multiplicative map. Thisproperty is a consequence of the characterizationgiven above of the determinant as the unique n-

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known as the special linear group. More generally,the word "special" indicates the subgroup of anothermatrix group of matrices of determinant one.Examples include the special orthogonal group

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