In mathematics, the determinant is a scalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the study of square matrices and linear transformations represented by them.
The determinant of a matrix is commonly denoted , , or . For square matrices of dimensions two and three, the determinant is explicitly described as
There are several equivalent ways to define the determinant of an matrix. A simple and conceptual one is that it 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 the 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. In the common case where the entries belong to a field, a matrix is invertible if and only if the determinant is nonzero. The adjugate matrix provides an explicit expression of the inverse matrix.
A consequence of the multiplicative property is that two similar matrices have the same determinant, and therefore the determinant of a linear transformation is well defined. For real linear transformations, the value of the determinant is the scale factor by which the transformation alters every volume in the space (positive if the orientation is preserved, negative if the orientation is reversed, zero if the transformation is not bijective).
The determinant is linear with respect to each row and each column of the matrix and is therefore a multilinear function. This means that it is a linear combination of the entries of each row or column of the matrix. Laplace expansion expresses the coefficients of this linear combination as determinants (of one dimension less), called cofactors. Laplace expansion is useful for computing determinants when the rows or columns are very sparse, typically when there are only one or two nonzero entries in a row or column.
Determinants can be efficiently computed with Gaussian elimination and singular value decomposition. Determinants can also be theoretically computed by means of the Leibniz formula or Laplace expansion, but this is inefficient except in special cases.
Determinants occur throughout mathematics. Cramer's rule expresses the solution of a system of linear equations in terms of determinants. The characteristic polynomial of a square matrix, whose roots are the eigenvalues, is defined as a determinant with polynomial entries. In geometry, determinants are used to define or compute areas and volumes, as well as space orientation and colinearity. In analysis, the Jacobian determinant is used for change of variables in multiple integrals and its non-nullity is one of the hypotheses of the implicit function theorem in several variables. In algebraic geometry, the Hessian determinant allows finding inflexion points of an algebraic plane curve.
Low dimension
Dimension < 2
Square matrices with less than two rows and two columns are rarely considered per se, but may occur when working with matrices where is a unspecified natural number.
A matrix is the empty matrix with 0 row, 0 column and no entries. The usual conventions for the empty set and the empty product imply that the determinant of the empty matrix is 1.
A matrix has a single entry, and its determinant equals this entry:
An example of the coherency of these conventions is the formula for the inverse matrix of a matrix with a nonzero determinant : the inverse is the multiplication by of the adjugate matrix of , which is a matrix whose entries are, up to the sign, determinants of submatrices of . In the case , the single entry of the adjugate matrix is the determinant of the empty matrix, which equals 1. So,
which is what is expected for matrices.
Two by two matrices The determinant of a matrix is
This simple form allows verifying by direct computation the properties listed in .
Parallelogram area
One of the main uses of determinants is the computation of the area of the parallelogram defined by two vectors in a Euclidean plane.
In fact, the two column vectors and of the matrix define a parallelogram with vertices , , , and . The area of this parallogram is the absolute value of the determinant of the matrix. The sign of the determinant determines the relative orientation of the vectors: the determinant is positive if one passes by a counterclockwise rotation from the direction of the vector to that of, and negative if a clockwise rotation is needed (if the vectors are aligned, the determinant is zero). So, the determinant is the signed area of the parallelogram.
The proof that the determinant is the signed area of the parallelogram can be done either with trigonometry or by dissection and rearrangement, as shown in the figure.
3 × 3 matrices By the Leibniz formula for determinants, the determinant of a matrix is
The rule of Sarrus is a mnemonic for the expanded form of this determinant: 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. This scheme for calculating the determinant of a matrix does not carry over into higher dimensions.
Definitions
The determinant of a square matrix is a scalar uniquely associated to it. Commonly, matrix entries and determinants are real or complex numbers, but, more generally, matrix entries may belong to any commutative ring, in which case the scalars and the determinant belong to this ring. If
is a square matrix with rows and columns, its determinant is denoted , ,
or
There are various equivalent ways to define the determinant of a square matrix , and it depends on authors which definition is chosen as the primary one, as well as the level of generality. In this article, the definition by the multiplicative property has been chosen as the primary one, because it is simple and does not require specific background.
In this section, the scalars (matrix entries and determinants) are assumed to belong to a field, such as the real or complex numbers. The multiplicative and elementary-transformation characterizations remain valid over an integral domain, but their equivalence requires a different argument, using fraction-free elimination and cancellation rather than ordinary Gaussian elimination. That argument is not given here.
By multiplicative property The determinant is the unique function from the square matrices to the scalars that maps a product of matrices of the same size to the product of their determinants, and maps every triangular matrix to the product of their diagonal entries. That is,
and
Such a definition requires the proof that such a function exists and is unique. This will be a byproduct of the proof of the equivalence with other definitions.
By elementary transformations
An elementary row transformation is a transformation of a matrix of one of the following types, where denotes the th row of the matrix: Multiplying one row by a scalar: Adding to a row a scalar multiple of another row: Exchanging two rows: Elementary column transformations are defined similarly by replacing "row" with "column". Transformations of type 3 may be obtained as a succession of thansformations of types 1 and 2: the exchange of rows and may be done by the following succession of transformations, where denotes the current th row (not necessarily the original one):
This leads to another definition of determinants: The determinant is the unique function of the entries of a square matrix such that the determinant of the identity matrix is 1, the transformations of type 1 multiply the determinant by the involved scalar, and the transformations of type 2 do not change the determinant; it follows that the transformations of type 3 multiply the determinant by –1.
For proving that this definition is equivalent to the preceding one, one has to interpret elementary transformations in terms of elementary matrices. In fact, an elementary transformation amounts to left-multiply the matrix by the diagonal matrix with all diagonal elements equal to except the th that equals . An elementary transformation amounts to left-multiplication of the matrix by a triangular matrix in which all diagonal entries are equal to , the entry in the th row and the th column is equal to , and all other entries are zero.
A determinant function that satisfies the definition by multiplicative properties satisfy the definition by elementary transformations since the identity matrix is a triangular matrix and the other conditions of the latter definition are the restriction of the multiplicative property to the case where the left factor is an elementary matrix.
Conversely, a determinant function that satisfies the definition by elementary transformations satisfies also the other definition. Over a field, Gaussian elimination allows proving that every matrix can be expressed as a product of elementary matrices (provided that zero is allowed as a scalar for the elementary transformations of type 1). So, for proving , it suffices to express as a product of elementary matrices and apply associative property of matrix product.
Uniqueness of the determinant results also from Gaussian elimination. Indeed, over a field one may transform, by means of elementary transformations, any matrix into a row echelon form, which is a specific triangular matrix. This allows computing the determinant. However, it does not by itself establish that the value obtained is independent of the elimination path or of the elementary matrices, so it does not prove the existence of the determinant. Instead, what it shows is that if two determinant functions are given, then they agree on every such path and therefore are identical. So it remains only to construct a function that satisfies the properties defining the determinant, a task taken up below.
By Leibniz formula
The Leibniz formula is one of the most common definitions of determinants. It is an explicit expression of a determinant as a multivariate polynomial function of the matrix entries. It also makes sense in a general commutative ring, and does not require it to be a field or integral domain. The proof that it satisfies the preceding definitions implies the required existence proof. Also, per uniqueness in the special case of an integral domain, this proves also the equivalence with the other definitions.
To write the Leibniz formula, one needs permutations and their signatures. That is, the determinant of a matrix
is
where runs over all permutations of .
When is the empty matrix, the usual conventions for the empty set and the empty product give . If ,one has . Otherwise, the above formula is often expanded as
The Levi-Civita symbol allows a variant of the Leibniz formula that is useful in some contexts. It is denoted and defined on the -tuples of integers in as if two of the integers are equal, and otherwise as the signature of the permutation formed by the n-tuple. With the Levi-Civita symbol, the Leibniz formula becomes
where the sum is taken over all -tuples of integers in
To prove that the Leibniz formula satisfies the definition of the preceding sections, one proves first that it is an alternating multilinear form that takes the value 1 on the identity matrix (see next section): multilinearity results from the fact that the Leibniz formula is a homogeneous polynomial of degree one in the entries of each column of the matrix; the fact that the form is alternating and takes the value 1 on the identity matrix result from basic properties of permutations. Then, it suffices to applies the results of the next section.
As an alternating multilinear form