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# By the name of Allah

The Determinant

## Edit by: Ahmed osman

Determinant:
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well. The determinant provides important information when the matrix is that of the coefficients of a system of linear equations, or when it corresponds to a linear transformation of a vector space: in the first case the system has a unique solution if and only if the determinant is nonzero, in the second case that same condition means that the transformation has an inverse operation. A geometric interpretation can be given to the value of the determinant of a square matrix with real entries:

the absolute value of the determinant gives the scale factor by which area or volume is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation. Thus a 2 2 matrix with determinant 2, when applied to a region of the plane with finite area, will transform that region into one with twice the area, while reversing its orientation. Determinants occur throughout mathematics. The use of determinants in calculus includes the Jacobian determinant in the substitution rule for integrals of functions of several variables. They are used to define the characteristic polynomial of a matrix that is an essential tool in eigenvalue problems in linear algebra. In some cases they are used just as a compact notation for expressions that would otherwise be unwieldy to write down.

The determinant of a matrix A is denoted det(A), det A, or |A|.[1] In the case where the matrix entries are written out in full, the determinant is denoted by surrounding the matrix entries by vertical bars instead 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 entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring. Thus for instance the determinant of a matrix with integer coefficients will be an integer, and the matrix has an inverse with integer coefficients if and only if this determinant is 1 or 1 (these being the

Contents
only inverti ble eleme nts of the intege rs). For square matrices with entries in a noncommutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

1 Definition
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## 2 Properties of the determinant

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2.1 Multiplicativity and matrix groups 2.2 Laplace's formula and the adjugate matrix 2.3 Sylvester's determinant theorem

## 3 Properties of the determinant in relation to other notions

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3.1 Relation to eigenvalues and trace 3.2 Cramer's rule 3.3 Block matrices 3.4 Derivative 4.1 Determinant of an endomorphism

## 4 Abstract algebraic aspects

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4.2 Exterior algebra 4.3 Square matrices over commutative rings and abstract properties

## 5 Generalizations and related notions

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5.1 Infinite matrices 5.2 Notions of determinant over noncommutative rings 5.3 Further variants 6.1 Decomposition methods 6.2 Further methods

6 Calculation
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7 History 8 Applications
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8.1 Linear independence 8.2 Orientation of a basis 8.3 Volume and Jacobian determinant 8.4 Vander monde determinant (alternant) 8.5 Circulates

] Defi nitio n

There are various ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns. Perhaps the most natural way is expressed in terms of the columns of the matrix. If we write an n-by-n matrix in terms of its column vectors

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

where b and c are scalars, v is any vector of size n and I is the identity matrix of size n. These properties state that the determinant is an alternating multilinear function of the columns, and they suffice to uniquely calculate the determinant of any square matrix. Provided the underlying scalars form a field (more generally, a commutative ring with unity), the definition below shows that such a function exists, and it can be shown to be unique.[2] Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is -1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries. This expression grows rapidly with the size of the matrix (an n-by-n matrix contributes n! terms), so it will first

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

The entries can be numbers or expressions (as happens when the determinant is used to define a characteristic polynomial); the definition of the determinant depends only on the fact that they can be added and multiplied together in a commutative manner.

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

##  2-by-2 matrices

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

If the matrix entries are real numbers, the matrix A can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of

## A, and one that maps them to the columns of A. In

either case, the images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at (0,0), (a,b), (a + c, b + d), and (c,d), as shown in the accompanying diagram. The absolute value of is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by A. (The parallelogram formed by the columns of A is in general a different parallelogram, but since the

determinant is symmetric with respect to rows and columns, the area will be the same.) The absolute value of the determinant together with the sign becomes the oriented area of the parallelogram. The oriented area is the same as the usual area, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the identity matrix). Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by A. When the determinant is equal to one, the linear mapping defined by the matrix represents is equi-areal and orientation-preserving. 3-by-3 matrices

The volume of this Parallelepiped is the absolute value of the determinant of the matrix formed by the rows r1, r2, and r3. The determinant of a 33 matrix is defined by

The determinant of a 3x3 matrix can be calculated by its diagonals. The rule of Sarrus is a mnemonic for this formula: 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 in the illustration at the right. For example, the determinant of

## is calculated using this rule:

This scheme for calculating the determinant of a 33 matrix does not carry over into higher dimensions.

n-by-n matrices
The determinant of a matrix of arbitrary size can be defined by the Leibniz formula or the Laplace formula. The Leibniz formula for the determinant of an n-by-n matrix A is

Here the sum is computed over all permutations of the set {1, 2, ..., n}. A permutation is a function that reorders this set of integers. The position of the element i after the reordering is denoted i. For example, for n = 3, the original sequence 1, 2, 3 might be reordered to = [2, 3, 1], with 1 = 2, 2 = 3, and 3 = 1. The set of all such permutations (also known as the symmetric group on n elements) is denoted Sn. For each permutation , sgn() denotes

the signature of ; it is +1 for even and 1 for odd . Evenness or oddness can be defined as follows: the permutation is even (odd) if the new sequence can be obtained by an even number (odd, respectively) of switches of numbers. For example, starting from [1, 2, 3] (and starting with the convention that the signature sgn([1,2,3]) = +1) and switching the positions of 2 and 3 yields [1, 3, 2], with sgn([1,3,2]) = 1. Switching once more yields [3, 1, 2], with sgn([3,1,2]) = +1 again. Finally, after a total of three switches (an odd number), the resulting permutation is [3, 2, 1], with sgn([3,2,1]) = 1. Therefore [3, 2, 1] is an odd permutation. Similarly, the permutation [2, 3, 1] is even: [1, 2, 3] [2, 1, 3] [2, 3, 1], with an even number of switches. A permutation cannot be simultaneously even and odd, but sometimes it is convenient to accept nonpermutations: sequences with repeated or skipped

numbers, like [1, 2, 1]. In that case, the signature of any non-permutation is zero: sgn([1,2,1]) = 0. In any of the summands, the term

is notation for the product of the entries at positions (i, i), where i ranges from 1 to n:

## For example, the determinant of a 3 by 3 matrix A (n = 3) is

This agrees with the rule of Sarrus given in the previous section.

The formal extension to arbitrary dimensions was made by Tullio Levi-Civita, see (Levi-Civita symbol) using a pseudo-tensor symbol.  Levi-Civita symbol The determinant for an n-by-n matrix can be expressed in terms of the totally antisymmetric LeviCivita symbol as follows:

##  Properties of the determinant

The determinant has many properties. Some basic properties of determinants are: 1. The determinant of the nn identity matrix equals 1. 2. Viewing an nn matrix as being composed of n columns, the determinant is an n-linear function.

This means that if one column of a matrix A is written as a sum v + w of two column vectors, and all other columns are left unchanged, then the determinant of A is the sum determinants of the matrices obtained from A by replacing the column by v respectively by w (and a similar relation holds when writing a column as a scalar multiple of a column vector). 3. This n-linear function is an alternating form. This means that whenever two columns of a matrix are identical, or more generally some column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), its determinant is 0. These properties, which follow from the Leibniz formula, already completely characterize the

determinant; in other words the determinant is the unique function from nn matrices to scalars that is

## n-linear alternating in the columns, and takes the

value 1 for the identity matrix (this characterization holds even if scalars are taken in any given commutative ring). To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants are either 0 (by property 3) or else 1 (by properties 1 and 7 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear. For matrices over non-commutative rings, properties 1 and 2 are

incompatible for n 2,[3] so there is no good definition of the determinant in this setting. 4. A matrix and its transpose have the same determinant. This implies that properties for columns have their counterparts in terms of rows: 5. Viewing an nn matrix as being composed of n rows, the determinant is an n-linear function. 6. This n-linear function is an alternating form: whenever two rows of a matrix are identical, its determinant is 0. 7. Interchanging two columns of a matrix multiplies its determinant by 1. This follows from properties 2 and 3 (it is a general property of multilinear alternating maps). Iterating gives that more generally a permutation of the columns multiplies the determinant by the sign of the permutation. Similarly a permutation of the rows

multiplies the determinant by the sign of the permutation. 8. Adding a scalar multiple of one column to

## another column does not change the value of

the determinant. This is a consequence of properties 2 and 3: by property 2 the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0 by property 3. Similarly, adding a scalar multiple of one row to another row leaves the determinant unchanged. 9. If A is a triangular matrix, i.e. ai,j = 0 whenever i > j or, alternatively, whenever i < j, then its determinant equals the product of the diagonal entries:

While this can be deduced from earlier properties, it follows most easily directly from the Leibniz formula (or from the Laplace expansion), in which the identity permutation is the only one that gives a non-zero contribution. These properties can be used to facilitate the computation of determinants by simplifying the matrix to the point where the determinant can be determined immediately. Specifically, for matrices with coefficients in a field, properties 7 and 8 can be used to transform any matrix into a triangular matrix, whose determinant is given by property 9; this is essentially the method of Gaussian elimination.

## For example, the determinant of computed using the following matrices:

can be

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

Thus the determinant is a multiplicative map. This property is a consequence of the characterization given above of the determinant as the unique n-

linear alternating function of the columns with value 1 on the identity matrix, since the function Mn(K)

## K that maps M det(AM) can easily be seen to be

n-linear and alternating in the columns of M, and
takes the value det(A) at the identity. The formula can be generalized to (square) products of rectangular matrices, giving the Cauchy-Binet formula, which also provides an independent proof of the multiplicative property. The determinant det(A) of a matrix A is non-zero if and only if A is invertible or, yet another equivalent statement, if its rank equals the size of the matrix. If so, the determinant of the inverse matrix is given by

In particular, products and inverses of matrices with determinant one still have this property. Thus, the set of such matrices (of fixed size n) form a group

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