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

 \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.

Contents

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 aj 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 adbc 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

\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\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 σi. For example, for n = 3, the original sequence 1, 2, 3 might be reordered to S = [2, 3, 1], with S1 = 2, S2 = 3, S3 = 1. The set of all such permutations (also known as the symmetric group on n elements) is denoted Sn. 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, σi), 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:

  1. The determinant of the n×n identity matrix equals 1.
  2. 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).
  3. 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 ≥ 2,[3] so there is no good definition of the determinant in this setting.

  1. A matrix and its transpose have the same determinant. This implies that properties for columns have their counterparts in terms of rows:
  2. Viewing an n×n matrix as being composed of n rows, the determinant is an n-linear function.
  3. This n-linear function is an alternating form: whenever two rows of a matrix are identical, its determinant is 0.
  4. 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.
  5. 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.
  1. 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:
    \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 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

\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 Mi,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+jMi,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,

det(Im + AB) = det(In + BA),

where Im and In are the m-by-m and n-by-n identity matrices, respectively.

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

det(Im + cr) = 1 + rc.

More generally, for any invertible m-by-m matrix X,[4]

det(X + AB) = det(X)det(In + BX − 1A),

and

det(X + cr) = det(X)(1 + rX − 1c).

Properties of the determinant in relation to other notions

Relation to eigenvalues and trace

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 Ai 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 aj 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 Rn×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, ..., vn, 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(In) = 1. Here In is the identity matrix.

This fact also implies that any every other n-linear alternating function F: Mn(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) ≠ 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 : VV 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 v1v2 ∧ ... ∧ vn to v2v1v3 ∧ ... ∧ vn, say, also alters the parity.

For this reason, the highest non-zero exterior power Λn(V) is sometimes also called the determinant of V and similarly for more involved objects such as vector bundles or chain complexes of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms ΛkV with k < n.

Square matrices over commutative rings and abstract properties

The determinant of a matrix can be defined, for example using the Leibniz formula, for matrices with entries in any commutative ring. Briefly, a ring is a structure where addition, subtraction, and multiplication are defined. The commutativity requirement means that the product does not depend on the order of the two factors, i.e.,

r \cdot s = s \cdot r

is supposed to hold for all elements r and s of the ring. For example, the integers form a commutative ring.

Many[clarification needed] of the above statements and notions carry over mutatis mutandis to determinants of these more general matrices: the determinant is multiplicative in this more general situation, and Cramer's rule also holds. A square matrix over a commutative ring R is invertible if and only if its determinant is a unit in R, that is, an element having a (multiplicative) inverse. (If R is a field, this latter condition is equivalent to the determinant being nonzero, thus giving back the above characterization.) For example, a matrix A with entries in Z, the integers, is invertible (in the sense that the inverse matrix has again integer entries) if the determinant is +1 or −1. Such a matrix is called unimodular.

The determinant defines a mapping between

\mathrm{GL}_n(R) \rightarrow R^\times, \,

the group of invertible n×n matrices with entries in R and the multiplicative group of units in R. Since it respects the multiplication in both groups, this map is a group homomorphism. Secondly, given a ring homomorphism f: RS, there is a map GLn(R) → GLn(S) given by replacing all entries in R by their images under f. The determinant respects these maps, i.e., given a matrix A = (ai,j) with entries in R, the identity

f(\det((a_{i,j}))) = \det ((f(a_{i,j})))\,

holds. For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo m of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo m (the latter determinant being computed using modular arithmetic). In the more high-brow parlance of category theory, the determinant is a natural transformation between the two functors GLn and (⋅)×.[9] Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of algebraic groups, from the general linear group to the multiplicative group,

\det: \mathrm{GL}_n \rightarrow \mathbb G_m.\,

Generalizations and related notions

Infinite matrices

For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in Leibniz' formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. Functional analysis provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators.

The Fredholm determinant defines the determinant for operators known as trace class operators by an appropriate generalization of the formula

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

Another infinite-dimensional notion of determinant is the functional determinant.

Notions of determinant over non-commutative rings

For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants in a manner analogous to that for commutative rings. A meaning can be given to the Leibniz formula provided the order for the product is specified, and similarly for other ways to define the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, for instance the multiplicative property or the fact that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (if a bilinear form exists with a regular element of R as value on some pair of arguments, it can be used to show that all elements of R commute). Nevertheless various notions of non-commutative determinant have been formulated, which preserve some of the properties of determinants, notably quasideterminants and the Dieudonné determinant.

Further variants

Determinants of matrices in superrings (that is, Z/2-graded rings) are known as Berezinians or superdeterminants.[10]

The permanent of a matrix is defined as the determinant, except that the factors sgn(σ) occurring in Leibniz' rule are omitted. The immanant generalizes both by introducing a character of the symmetric group Sn in Leibniz' rule.

Calculation

Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications like checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques.[11] Nonetheless, explicitly calculating determinants is required in some situations, and different methods are available to do so.

Naive methods of implementing an algorithm to compute the determinant include using Leibniz' formula or Laplace's formula. Both these approaches are extremely inefficient for large matrices, though, since the number of required operations grows very quickly: it is of order n! (n factorial) for an n×n matrix M. For example, Leibniz' formula requires to calculate n! products. Therefore, more involved techniques have been developed for calculating determinants.

Decomposition methods

Given a matrix A, some methods compute its determinant by writing A as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the LU decomposition, Cholesky decomposition or the QR decomposition. These methods are of order O(n3), which is a significant improvement over O(n!)

The LU decomposition expresses A in terms of a lower triangular matrix L, an upper triangular matrix U and a permutation matrix P:

 A = PLU. \,

The determinants of L and U can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of P is just the sign ε of the corresponding permutation. The determinant of A is then

 \det(A) = \varepsilon \det(L)\cdot\det(U), \,

Moreover, the decomposition can be chosen such that L is a unitriangular matrix and therefore has determinant 1, in which case the formula further simplifies to

det(A) = εdet(U).

Further methods

If the determinant of A and the inverse of A have already been computed, the matrix determinant lemma allows to quickly calculate the determinant of A + uvT, where u and v are column vectors.

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[12] 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.

If two matrices of order n can be multiplied in time M(n), where M(n)≥na for some a>2, then the determinant can be computed in time O(M(n)).[13] This means, for example, that an O(n2.376) algorithm exists based on the Coppersmith–Winograd algorithm.

Algorithms can also be assessed according to their bit complexity, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the Gaussian elimination (or LU decomposition) methods is of order O(n3), but the bit length of intermediate values can become exponentially long.[14] The Bareiss Algorithm, on the other hand, is an exact-division method based on Sylvester's identity is also of order n3, but the bit complexity is roughly the bit size of the original entries in the matrix times n.[15]

History

Historically, determinants were considered without reference to matrices: originally, a determinant was defined as a property of a system of linear equations. 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 Chinese mathematics textbook The Nine Chapters on the Mathematical Art (九章算術, Chinese scholars, around the 3rd century BC). In Europe, two-by-two determinants were considered by Cardano at the end of the 16th century and larger ones by Leibniz.[16][17][18][19]

In Europe, Cramer (1750) added to the theory, treating the subject in relation to sets of equations. The recurrence law was first announced by Bézout (1764).

It was Vandermonde (1771) who first recognized determinants as independent functions.[16] Laplace (1772) [20][21] gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case. 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.

Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in the theory of numbers. 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. 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. On the same day (November 30, 1812) that Binet presented his paper to the Academy, Cauchy also presented one on the subject. (See Cauchy-Binet formula.) In this he used the word determinant in its present sense,[22][23] 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.[16][24] With him begins the theory in its generality.

The next important figure was Jacobi[17] (from 1827). 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. About the time of Jacobi's last memoirs, Sylvester (1839) and Cayley began their work.[25][26]

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. Of the text-books on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.

Axler in 1995 attacked determinant's place in Linear Algebra. He saw it as something to be derived from the core principles of Linear Algebra, not to be used to derive the core principles.[27]

Applications

Vandermonde determinant (alternant)

Third order

\left|
\begin{array}{ccc}
 1 & 1 & 1 \\
 x_1 & x_2 & x_3 \\
 x_1^2 & x_2^2 & x_3^2
\end{array}
\right|=\left(x_3-x_2\right)\left(x_3-x_1\right)\left(x_2-x_1\right).

In general, the nth-order Vandermonde determinant is [28]

\left|
\begin{array}{ccccc}
 1 & 1 & 1 & \cdots  & 1 \\
 x_1 & x_2 & x_3 & \cdots  & x_n \\
 x_1^2 & x_2^2 & x_3^2 & \cdots  & x_n^2 \\
 \vdots  & \vdots  & \vdots  & \ddots & \vdots  \\
 x_1^n & x_2^n & x_2^n & \cdots  & x_n^{n-1}
\end{array}
\right|=\prod _{1\leq i<j\leq n} \left(x_j-x_i\right),

where the right-hand side is the continued product of all the differences that can be formed from the n(n-1)/2 pairs of numbers taken from x1, x2, ... ,xn, with the order of the differences taken in the reversed order of the suffixes that are involved.

Circulants

Second order

\left|
\begin{array}{cc}
 x_1 & x_2 \\
 x_2 & x_1
\end{array}
\right|=\left(x_1+x_2\right)\left(x_1-x_2\right).

Third order

\left|
\begin{array}{ccc}
 x_1 & x_2 & x_3 \\
 x_3 & x_1 & x_2 \\
 x_2 & x_3 & x_1
\end{array}
\right|=\left(x_1+x_2+x_3\right)\left(x_1+\omega  x_2+\omega ^2x_3\right)\left(x_1+\omega ^2x_2+\omega  x_3\right),

where ω and ω2 are the complex cube roots of 1. In general, the nth-order circulant determinant is [28]

\left|
\begin{array}{ccccc}
 x_1 & x_2 & x_3 & \cdots  & x_n \\
 x_n & x_1 & x_2 & \cdots  & x_{n-1} \\
 x_{n-1} & x_n & x_1 & \cdots  & x_{n-2} \\
 \vdots  & \vdots  & \vdots  & \ddots & \vdots  \\
 x_2 & x_3 & x_4 & \cdots  & x_1
\end{array}
\right|=\prod _{j=1}^n \left(x_1+x_2\omega _j+x_3\omega _j^2+\ldots +x_n\omega _j^{n-1}\right),

where ωj is an nth root of 1.

Linear independence

As mentioned above, the determinant of a matrix (with real or complex entries, say) is zero if and only if the column vectors of the matrix are linearly dependent. Thus, determinants can be used to characterize linearly dependent vectors. For example, given two vectors v1, v2 in R3, a third vector v3 lies in the plane spanned by the former two vectors exactly if the determinant of the 3-by-3 matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given n functions f1(x), ..., fn(x) (supposed to be n−1 times differentiable), the Wronskian is defined to be


W(f_1, \ldots, f_n) (x)=
\begin{vmatrix} 
f_1(x) & f_2(x) & \cdots & f_n(x) \\
f_1'(x) & f_2'(x) & \cdots & f_n' (x)\\
\vdots & \vdots & \ddots & \vdots \\
f_1^{(n-1)}(x)& f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x)
\end{vmatrix}.

It is non-zero (for some x) in a specified interval if and only if the given functions and all their derivatives up to order n−1 are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence.

Orientation of a basis

The determinant can be thought of as assigning a number to every sequence of n in Rn, by using the square matrix whose columns are the given vectors. For instance, an orthogonal matrix with entries in Rn represents an orthonormal basis in Euclidean space. The determinant of such a matrix determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. Namely, if the determinant is +1, the basis has the same orientation. If it is −1, the basis has the opposite orientation.

More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal 2×2 or 3×3 matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.

Volume and Jacobian determinant

As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if f: Rn−1Rn is the linear map represented by the matrix A, and S is any measurable subset of Rn, then the volume of f(S) is given by |det(A)| times the volume of S. More generally, if the linear map f: Rn−1Rm is represented by the m-by-n matrix A, then the m-dimensional volume of f(S) is given by:

\operatorname {volume} (f(S)) = \sqrt{\det(A^\mathrm{T} A)} \times \operatorname{volume}(S).

By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. The volume of any tetrahedron, given its vertices a, b, c, and d, is (1/6)·|det(a − bb − c, c − d)|, or any other combination of pairs of vertices that would form a spanning tree over the vertices.

For a general differentiable function, much of the above carries over by considering the Jacobian matrix of f. For

f: \mathbf R^n \rightarrow \mathbf R^n,

the Jacobian is the n-by-n matrix whose entries are given by

D(f) = \left (\frac {\partial f_i}{\partial x_j} \right )_{1 \leq i, j \leq n}. \,

Its determinant, the Jacobian determinant appears in the higher-dimensional version of integration by substitution: for suitable functions f and an open subset U of R'n (the domain of f), the integral over f(U) of some other function φ: RnRm is given by

 \int_{f(U)} \varphi(\mathbf{v})\, d \mathbf{v} = \int_U \phi(f(\mathbf{u})) \left|\det(\operatorname{D}f)(\mathbf{u})\right| \,d \mathbf{u}.

The Jacobian also occurs in the inverse function theorem.

See also

Notes

  1. ^ Poole, David (2006), Linear Algebra: A Modern Introduction, Thomson Brooks/Cole, p. 262, ISBN 0-534-99845-3 
  2. ^ Serge Lang, Linear Algebra, 2nd Edition, Addison-Wesley, 1971, pp 173, 191.
  3. ^ In a non-commutative setting left-linearity (compatibility with left-multiplication by scalars) should be distinguished from right-linearity. Assuming linearity in the columns is taken to be left-linearity, one would have, for non-commuting scalars a, b:
    ab = ab \left|\begin{matrix}1&0\\0&1\end{matrix} \right| = a \left|\begin{matrix}1&0\\0&b\end{matrix} \right| = \left|\begin{matrix}a&0\\0&b\end{matrix} \right| = b \left|\begin{matrix}a&0\\0&1\end{matrix} \right| = ba \left|\begin{matrix}1&0\\0&1\end{matrix} \right|= ba,
    a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring.
  4. ^ Proofs can be found in http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html
  5. ^ Ken Habgood, Itamar Arel, A condensation-based application of Cramerʼs rule for solving large-scale linear systems, Journal of Discrete Algorithms, Available online 1 July 2011, ISSN 1570-8667, 10.1016/j.jda.2011.06.007.
  6. ^ These identities were taken http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html
  7. ^ Proofs are given at http://www.mth.kcl.ac.uk/~jrs/gazette/blocks.pdf
  8. ^ Roger Godement, Cours d'Algèbre, seconde édition, Hermann (1966), §23, Théorème 5, p. 303
  9. ^ Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 ((2nd ed.) ed.), Springer-Verlag, ISBN 0-387-98403-8 
  10. ^ Varadarajan, V. S (2004), Supersymmetry for mathematicians: An introduction, ISBN 9780821835746, http://books.google.com/?id=sZ1-G4hQgIIC&pg=PA116&dq=Berezinian#v=onepage&q=Berezinian&f=false. 
  11. ^ L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997). e.g. in Lecture 1: "... we mention that the determinant, though a convenient notion theoretically, rarely finds a useful role in numerical algorithms."
  12. ^ http://page.inf.fu-berlin.de/~rote/Papers/pdf/Division-free+algorithms.pdf
  13. ^ J.R. Bunch and J.E. Hopcroft, Triangular factorization and inversion by fast matrix multiplication, Mathematics of Computation, 28 (1974) 231–236.
  14. ^ Fang, Xin Gui; Havas, George (1997). "On the worst-case complexity of integer Gaussian elimination". Proceedings of the 1997 international symposium on Symbolic and algebraic computation. ISSAC '97. Kihei, Maui, Hawaii, United States: ACM. pp. 28–31. doi:http://doi.acm.org/10.1145/258726.258740. ISBN 0-89791-875-4. http://perso.ens-lyon.fr/gilles.villard/BIBLIOGRAPHIE/PDF/ft_gateway.cfm.pdf. 
  15. ^ Bareiss, Erwin (1968), "Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination", Mathematics of computation 22 (102): 565–578, http://www.ams.org/journals/mcom/1968-22-103/S0025-5718-1968-0226829-0/S0025-5718-1968-0226829-0.pdf 
  16. ^ a b c Campbell, H: "Linear Algebra With Applications", pages 111-112. Appleton Century Crofts, 1971
  17. ^ a b Eves, H: "An Introduction to the History of Mathematics", pages 405, 493–494, Saunders College Publishing, 1990.
  18. ^ A Brief History of Linear Algebra and Matrix Theory : http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html
  19. ^ Cajori, F. A History of Mathematics p. 80
  20. ^ Expansion of determinants in terms of minors: Laplace, Pierre-Simon (de) "Researches sur le calcul intégral et sur le systéme du monde," Histoire de l'Académie Royale des Sciences (Paris), seconde partie, pages 267-376 (1772).
  21. ^ Muir, Sir Thomas, The Theory of Determinants in the historical Order of Development [London, England: Macmillan and Co., Ltd., 1906].
  22. ^ The first use of the word "determinant" in the modern sense appeared in: Cauchy, Augustin-Louis “Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et des signes contraires par suite des transpositions operées entre les variables qu'elles renferment," which was first read at the Institute de France in Paris on November 30, 1812, and which was subsequently published in the Journal de l'Ecole Polytechnique, Cahier 17, Tome 10, pages 29-112 (1815).
  23. ^ Origins of mathematical terms: http://jeff560.tripod.com/d.html
  24. ^ History of matrices and determinants: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html
  25. ^ The first use of vertical lines to denote a determinant appeared in: Cayley, Arthur "On a theorem in the geometry of position," Cambridge Mathematical Journal, vol. 2, pages 267-271 (1841).
  26. ^ History of matrix notation: http://jeff560.tripod.com/matrices.html
  27. ^ Down with Determinants: http://www.axler.net/DwD.html
  28. ^ a b Gradshteyn, I. S., I. M. Ryzhik: "Table of Integrals, Series, and Products", 14.31, Elsevier, 2007.

References

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