Determinant

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This article is about determinants in mathematics. For determinants in epidemiology, see Risk factor.

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

$\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$ is written $\begin{vmatrix} a & b & c\\d & e & f\\g & h & i \end{vmatrix}$ and has the value $aei+bfg+cdh-ceg-bdi-afh\,$ .

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 only invertible elements of the integers). For square matrices with entries in a non-commutative 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
2 Properties of the determinant
3 Properties of the determinant in relation to other notions
4 Abstract algebraic aspects
5 Generalizations and related notions
6 Calculation
- 6.1 Decomposition methods
- 6.2 Further methods
7 History
8 Applications
9 See also
10 Notes
11 References
12 External links

[ Definition

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

$A = \begin{bmatrix} a_1, & a_2, & \ldots, & a_n \end{bmatrix}$

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

$\det\begin{bmatrix} a_1, & \ldots, & b a_j + c v, & \ldots, a_n \end{bmatrix} = b \det(A) + c \det\begin{bmatrix} a_1, & \ldots, & v, & \ldots, a_n \end{bmatrix}$

$\det\begin{bmatrix} a_1, & \ldots, & a_j, & a_{j+1}, & \ldots, a_n \end{bmatrix} = -\det\begin{bmatrix} a_1, & \ldots, & a_{j+1}, & a_j, & \ldots, a_n \end{bmatrix}$

$\det(I) = 1 \,$

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

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

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:

$\begin{vmatrix} a_{1,1} & a_{1,2} & \dots & a_{1,n} \\ a_{2,1} & a_{2,2} & \dots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & a_{n,2} & \dots & a_{n,n} \end{vmatrix}.\,$

[ 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 2×2 matrix is defined by

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

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 $a d - b c$ 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 3×3 matrix is defined by

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

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

$A = \begin{bmatrix}-2&2&3\\ -1& 1& 3\\ 2 &0 &-1\end{bmatrix}$

is calculated using this rule:

$\det(A)\,$	$=\,$	$((-2) \cdot 1 \cdot (-1)) + (2 \cdot 3 \cdot 2 ) + (3 \cdot (-1) \cdot 0)$
		$-\,(2 \cdot 1 \cdot 3) - (0 \cdot 3 \cdot (-2) ) - ((-1) \cdot (-1) \cdot 2)$
	$=\,$	$2 + 12 + 0 - 6 - 0 - 2 = 6.\,$

This scheme for calculating the determinant of a 3×3 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

$\det(A) = \sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i=1}^n A_{i,\sigma_i}.\$

Here the sum is computed over all permutations σ 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 σ is denoted σ_i. For example, for n = 3, the original sequence 1, 2, 3 might be reordered to σ = [2, 3, 1], with σ₁ = 2, σ₂ = 3, and σ₃ = 1. The set of all such permutations (also known as the symmetric group on n elements) is denoted S_n. For each permutation σ, sgn(σ) denotes the signature of σ; it is +1 for even σ and −1 for odd σ. 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 non-permutations: 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 $n!$ summands, the term

$\prod_{i=1}^n A_{i, \sigma_i}\$

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

$A_{1, \sigma_1} \cdot A_{2, \sigma_2} \cdots A_{n, \sigma_n}.\$

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

$\begin{align} \sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i=1}^n A_{i,\sigma_i} &=\sgn([1,2,3]) \prod_{i=1}^n A_{i,[1,2,3]_i} + \sgn([1,3,2]) \prod_{i=1}^n A_{i,[1,3,2]_i} + \sgn([2,1,3]) \prod_{i=1}^n A_{i,[2,1,3]_i} \\ &+ \sgn([2,3,1]) \prod_{i=1}^n A_{i,[2,3,1]_i} + \sgn([3,1,2]) \prod_{i=1}^n A_{i,[3,1,2]_i} + \sgn([3,2,1]) \prod_{i=1}^n A_{i,[3,2,1]_i} \\ &=\prod_{i=1}^n A_{i,[1,2,3]_i} - \prod_{i=1}^n A_{i,[1,3,2]_i} - \prod_{i=1}^n A_{i,[2,1,3]_i} + \prod_{i=1}^n A_{i,[2,3,1]_i} + \prod_{i=1}^n A_{i,[3,1,2]_i} - \prod_{i=1}^n A_{i,[3,2,1]_i} \\ &=A_{1,1}A_{2,2}A_{3,3}-A_{1,1}A_{2,3}A_{3,2}-A_{1,2}A_{2,1}A_{3,3}+A_{1,2}A_{2,3}A_{3,1}+A_{1,3}A_{2,1}A_{3,2}-A_{1,3}A_{2,2}A_{3,1}. \end{align}$

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 Levi-Civita symbol as follows:

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

[ Properties of the determinant

The determinant has many properties. Some basic properties of determinants are:

The determinant of the n×n identity matrix equals 1.
Viewing an n×n 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).
This n-linear function is an alternating form. This means that whenever two columns of a matrix are identical, 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 n×n 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 $n \geq 2$ ,^[3] so there is no good definition of the determinant in this setting.

A matrix and its transpose have the same determinant. This implies that properties for columns have their counterparts in terms of rows:
Viewing an n×n matrix as being composed of n rows, the determinant is an n-linear function.
This n-linear function is an alternating form: whenever two rows of a matrix are identical, its determinant is 0.
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.
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.

If A is a triangular matrix, i.e. a_i,j = 0 whenever i > j or, alternatively, whenever i < j, then its determinant equals the product of the diagonal entries:

$\det(A) = a_{1,1} a_{2,2} \cdots a_{n,n} = \prod_{i=1}^n a_{i,i}.$

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 $A = \begin{bmatrix}-2&2&-3\\ -1& 1& 3\\ 2 &0 &-1\end{bmatrix}$ can be computed using the following matrices:

$B = \begin{bmatrix}-2&2&-3\\ 0 & 0 & 4.5\\ 2 &0 &-1\end{bmatrix}, C = \begin{bmatrix}-2&2&-3\\ 0 & 0 & 4.5\\ 0 & 2 &-4\end{bmatrix}, D = \begin{bmatrix}-2&2&-3\\ 0 & 2 &-4\\ 0 & 0 & 4.5 \end{bmatrix}.$

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) = +18.

[ Multiplicativity and matrix groups

The determinant of a matrix product of square matrices equals the product of their determinants:

$\det(AB) = \det (A) \det (B).\$

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 $M n (K) \to K$ that maps $M \mapsto det(AM)$ 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

$\det (A^{-1}) = \frac 1 {\det (A)}.$

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 (which if n is 2 or 3 consists of all rotation matrices), and the special unitary group.

[ Laplace's formula and the adjugate matrix

Laplace's formula expresses the determinant of a matrix in terms of its minors. The minor M_i,j is defined to be the determinant of the (n−1)×(n−1)-matrix that results from A by removing the i-th row and the j-th column. The expression (−1)^i+jM_i,j is known as cofactor. The determinant of A is given by

$\det(A) = \sum_{j=1}^n (-1)^{i+j} a_{i,j} M_{i,j} = \sum_{i=1}^n (-1)^{i+j} a_{i,j} M_{i,j}.$

Calculating det(A) by means of that formula is referred to as expanding the determinant along a row or column. For the example 3-by-3 matrix $A = \begin{bmatrix}-2&2&-3\\ -1& 1& 3\\ 2 &0 &-1\end{bmatrix}$ , Laplace expansion along the second column (j = 2, the sum runs over i) yields:

$\det(A)\,$	$=\,$	$(-1)^{1+2}\cdot 2 \cdot \det \begin{bmatrix}-1&3\\ 2 &-1\end{bmatrix} + (-1)^{2+2}\cdot 1 \cdot \det \begin{bmatrix}-2&-3\\ 2&-1\end{bmatrix} + (-1)^{3+2}\cdot 0 \cdot \det \begin{bmatrix}-2&-3\\ -1&3\end{bmatrix}$
	$=\,$	$(-2)\cdot((-1)\cdot(-1)-2\cdot3)+1\cdot((-2)\cdot(-1)-2\cdot(-3))$
	$=\,$	$(-2)\cdot(-5)+8 = 18.\,$

However, Laplace expansion is efficient for small matrices only.

The adjugate matrix adj(A) is the transpose of the matrix consisting of the cofactors, i.e.,

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

[ Sylvester's determinant theorem

Sylvester's determinant theorem states that for A, an m-by-n matrix, and B, an n-by-m matrix (so that A and B have dimensions allowing them to be multiplied in either order):

det(I m + A B) = det(I n + B A)

where $I m$ and $I n$ are the m-by-m and n-by-n identity matrices, respectively.

From this general result several consequences follow.

(a) For the case of column vector c and row vector r, each with m components, the formula allows quick calculation of the determinant of a matrix that differs from the identity matrix by a matrix of rank 1:

det(I m + c r) = 1 + r c

(b) More generally,^[4] for any invertible m-by-m matrix X,

det(X + A B) = det(X)det(I n + B X - 1 A)

[ Properties of the determinant in relation to other notions

[ Relation to eigenvalues and trace

Main article: Eigenvalues and eigenvectors

Determinants can be used to find the eigenvalues of the matrix A: they are the solutions of the characteristic equation

$\det(A - xI) = 0, \,$

where I is the identity matrix of the same dimension as A. Conversely, det(A) is the product of the eigenvalues of A, counted with their algebraic multiplicities. The product of all non-zero eigenvalues is referred to as pseudo-determinant.

An Hermitian matrix is positive definite if all its eigenvalues are positive. Sylvester's criterion asserts that this is equivalent to the determinants of the submatrices

$A_k := \begin{bmatrix} a_{1,1} & a_{1,2} & \dots & a_{1,k} \\ a_{2,1} & a_{2,2} & \dots & a_{2,k} \\ \vdots & \vdots & \ddots & \vdots \\ a_{k,1} & a_{k,2} & \dots & a_{k,k} \end{bmatrix}$

being positive, for all k between 1 and n.

The trace tr(A) is by definition the sum of the diagonal entries of A and also equals the sum of the eigenvalues. Thus, for complex matrices A,

$\det(\exp(A)) = \exp(\mathrm{tr}(A))\,$

or, for real matrices A,

$\mathrm{tr}(A) = \log(\det(\exp(A))). \,$

Here exp(A) denotes the matrix exponential of A, because every eigenvalue λ of A corresponds to the eigenvalue exp(λ) of exp(A). In particular, given any logarithm of A, that is, any matrix L satisfying

$\exp(L) = A\,$

the determinant of A is given by

$\det(A) = \exp(\mathrm{tr}(L)). \,$

For example, for n = 2 and n = 3, respectively,

$\det(A) = (\mathrm{tr}(A)^2 - \mathrm{tr}(A^2))/2, \,$

$\det(A) = (\mathrm{tr}(A)^3 - 3 \mathrm{tr}(A) \mathrm{tr}(A^2) + 2 \mathrm{tr}(A^3))/6. \,$

These formulae are closely related to Newton's identities.

A generalization of the above identities can be obtained from the following Taylor series expansion of the determinant:

$\begin{align} \det(\mathsf{I} + \mathsf{A}) = \sum_{k=0}^{\infty} \frac{1}{k!} \left( - \sum_{j=1}^{\infty} \frac{(-1)^j}{j}\mathrm{tr}(\mathsf{A}^j) \right) ^k\, , \end{align}$

where I is the identity matrix.

[ Cramer's rule

For a matrix equation

$Ax = b\,$

the solution is given by Cramer's rule:

$x_i = \frac{\det(A_i)}{\det(A)} \qquad i = 1, \ldots, n \,$

where A_i is the matrix formed by replacing the i-th column of A by the column vector b. This follows immediately by column expansion of the determinant, i.e.

$\det(A_i) = \det\begin{bmatrix}a_1, & \ldots, & b, & \ldots, & a_n\end{bmatrix} = \sum_{j=1}^n x_j\det\begin{bmatrix}a_1, & \ldots, a_{i-1}, & a_j, & a_{i+1}, & \ldots, & a_n \end{bmatrix} = x_i \det(A)$

where the vectors $a j$ are the columns of A. The rule is also implied by the identity

$A\, \mathrm{adj}(A) = \mathrm{adj}(A)\, A = \det(A)\, I_n.$

It has recently been shown that Cramer's rule can be implemented in O(n³) time,^[5] which is comparable to more common methods of solving systems of linear equations, such as LU, QR, or singular value decomposition.

[ Block matrices

Suppose A, B, C, and D are n×n-, n×m-, m×n-, and m×m-matrices, respectively. Then

$\det\begin{pmatrix}A& 0\\ C& D\end{pmatrix} = \det\begin{pmatrix}A& B\\ 0& D\end{pmatrix} = \det(A) \det(D) .$

This can be seen from the Leibniz formula or by induction on n. When A is invertible, employing the following identity

$\begin{pmatrix}A& B\\ C& D\end{pmatrix} = \begin{pmatrix}A& 0\\ C& I\end{pmatrix} \begin{pmatrix}I& A^{-1} B\\ 0& D - C A^{-1} B\end{pmatrix}$

leads to

$\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} = \det(A) \det(D - C A^{-1} B) .$

When D is invertible, a similar identity with $det(D)$ factored out can be derived analogously,^[6] that is,

$\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} = \det(D) \det(A - B D^{-1} C) .$

When the blocks are square matrices of the same order further formulas hold. For example, if C and D commute (i.e., CD = DC), then the following formula comparable to the determinant of a 2-by-2 matrix holds:^[7]

$\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} = \det(AD - BC).$

[ Derivative

By definition, e.g., using the Leibniz formula, the determinant of real (or analogously for complex) square matrices is a polynomial function from R^n×n to R. As such it is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:

$\frac{\mathrm{d} \det(A)}{\mathrm{d} \alpha} = \operatorname{tr}\left(\operatorname{adj}(A) \frac{\mathrm{d} A}{\mathrm{d} \alpha}\right).$

where adj(A) denotes the adjugate of A. In particular, if A is invertible, we have

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

Expressed in terms of the entries of A, these are

$\frac{\partial \det(A)}{\partial A_{ij}}= \operatorname{adj}(A)_{ji}= \det(A)(A^{-1})_{ji}.$

Yet another equivalent formulation is

$\det(A + \epsilon X) - \det(A) = \operatorname{tr}(\operatorname{adj}(A) X) \epsilon + O(\epsilon^2) = \det(A) \operatorname{tr}(A^{-1} X) \epsilon + O(\epsilon^2)$ ,

using big O notation. The special case where $A = I$ , the identity matrix, yields

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

This identity is used in describing the tangent space of certain matrix Lie groups.

If the matrix A is written as $A = \begin{bmatrix}\mathbf{a} & \mathbf{b} & \mathbf{c}\end{bmatrix}$ where a, b, c are vectors, then the gradient over one of the three vectors may be written as the cross product of the other two:

$\begin{align} \nabla_\mathbf{a}\det(A) &= \mathbf{b} \times \mathbf{c} \\ \nabla_\mathbf{b}\det(A) &= \mathbf{c} \times \mathbf{a} \\ \nabla_\mathbf{c}\det(A) &= \mathbf{a} \times \mathbf{b}. \end{align}$

[ Abstract algebraic aspects

[ Determinant of an endomorphism

The above identities concerning the determinant of a products and inverses of matrices imply that similar matrices have the same determinant: two matrices A and B are similar, if there exists an invertible matrix X such that A = X⁻¹BX. Indeed, repeatedly applying the above identities yields

$\det(A) = \det(X)^{-1} \det(BX) = \det(X)^{-1} \det(B)\det(X) = \det(B) \det(X)^{-1} \det(X) = \det(B).\$

The determinant is therefore also called a similarity invariant. The determinant of a linear transformation

$T : V \rightarrow V\,$

for some finite dimensional vector space V is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of basis in V. By the similarity invariance, this determinant is independent of the choice of the basis for V and therefore only depends on the endomorphism T.

[ Exterior algebra

The determinant can also be characterized as the unique function

$D: M_n(K) \to K\,$

from the set of all n-by-n matrices with entries in a field K to this field satisfying the following three properties: first, D is an n-linear function: considering all but one column of A fixed, the determinant is linear in the remaining column, that is

$D (v_1, \dots, v_{i-1}, a v_i + b w, v_{i+1}, \dots, v_n) = a D (v_1, \dots, v_{i-1}, v_i, v_{i+1}, \dots, v_n) + b D (v_1, \dots, v_{i-1}, w, v_{i+1}, \dots, v_n)\,$

for any column vectors v₁, ..., v_n, and w and any scalars (elements of K) a and b. Second, D is an alternating function: for any matrix A with two identical columns D(A) = 0. Finally, D(I_n) = 1. Here I_n is the identity matrix.

This fact also implies that any every other n-linear alternating function F: M_n(K) → K satisfies

$F(M)=F(I)D(M).\$

The last part in fact follows from the preceding statement: one easily sees that if F is nonzero it satisfies $F (I) \neq 0$ , and function that associates $F (M)/ F (I)$ to $M$ satisfies all conditions of the theorem. The importance of stating this part is mainly that it remains valid^[8] if K is any commutative ring rather than a field, in which case the given argument does not apply.

The determinant of a linear transformation A : V → V of an n-dimensional vector space V can be formulated in a coordinate-free manner by considering the n-th exterior power ΛⁿV of V. A induces a linear map

$\Lambda^n A: \Lambda^n V \rightarrow \Lambda^n V$

$v_1 \wedge v_2 \wedge \dots \wedge v_n \mapsto A v_1 \wedge A v_2 \wedge \dots \wedge A v_n.$

As ΛⁿV is one-dimensional, the map ΛⁿA is given by multiplying with some scalar. This scalar coincides with the determinant of A, that is to say

$(\Lambda^n A)(v_1 \wedge \dots \wedge v_n) = \det(A) \cdot v_1 \wedge \dots \wedge v_n.$

This definition agrees with the more concrete coordinate-dependent definition. This follows from the characterization of the determinant given above. For example, switching two columns changes the parity of the determinant; likewise, permuting the vectors in the exterior product v₁ ∧ v₂ ∧ ... ∧ v_n to v₂ ∧ v₁ ∧ v₃ ∧ ... ∧ v_n, say, also alters the parity.

For this reason, the highest non-zero exterior power Λⁿ(V) is sometimes also called thee determinant of V and similarly for more involved objects such as vector bundles or Source: this wikipedia article, under CC-BY-SA.

Determinant

Determinant

Contents

[ Definition

[ 2-by-2 matrices

[ 3-by-3 matrices

[ n-by-n matrices

[ Levi-Civita symbol

[ Properties of the determinant

[ Multiplicativity and matrix groups

[ Laplace's formula and the adjugate matrix

[ Sylvester's determinant theorem

[ Properties of the determinant in relation to other notions

[ Relation to eigenvalues and trace

[ Cramer's rule

[ Block matrices

[ Derivative

[ Abstract algebraic aspects

[ Determinant of an endomorphism

[ Exterior algebra

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