Determinant

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 A is commonly denoted det(A), det A, or |A|. For n by n matrices with ⁠\(n\le 3\)⁠, the determinant is explicitly described as \[\begin{aligned} \det[a]&=a \qquad (n=1),\\[10pt] \det \begin{bmatrix} a & b\\c & d \end{bmatrix}&= ad-bc \qquad (n=2),\\[10pt] \det \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} &= aei + bfg + cdh - ceg - bdi - afh \qquad (n=3). \end{aligned}\]

There are several equivalent ways to define the determinant of an ⁠\(n\times n\)⁠ 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.

Condensed: the full section is in Wikipedia.

Square matrices with less than two rows and two columns are rarely considered per se, but may occur when working with n × n matrices where n is a unspecified natural number.

A 0 × 0 is the empty matrix with 0 row, 0 column and no entries. The useal conventions for the empty set and the empty product imply that the determinant of the empty matrix is 1.

A 1 × 1 matrix has a single entry, and its determinant equals this entry: \[\det [a] = a\]

An example of the coherency of these conventions is the formula for the inverse matrix of a n × n matrix A with a nonzero determinant ⁠\(d\)⁠: the inverse is the multiplication by ⁠\(1/d\)⁠ of the adjugate matrix of A, which is a n × n matrix whose entries are, up to the sign, determinants of ⁠\((n-1)\times (n-1)\)⁠ submatrices of A. In the case ⁠\(n=1\)⁠, the single entry of the adjugate matrix is the determinant of the empty matrix, which equals 1. So, \[[d]^{-1} = \tfrac 1d \,[1] = \left[ \tfrac 1d \right],\] which is what is expected for 1 × 1 matrices.

The determinant of a 2 × 2 matrix \(\begin{pmatrix} a & b \\c & d \end{pmatrix}\) is

\[\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc.\] This simple form allows verifying by direct computation the properties listed in § Properties.

By the Leibniz formula for determinants, the determinant of a 3 × 3 matrix is \[\begin{vmatrix} a&b&c\\d&e&f\\g&h&i \end{vmatrix} = aei + bfg + cdh - ceg - bdi - afh.\]

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 3 × 3 matrix does not carry over into higher dimensions.

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 \[A = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{bmatrix}\] is a square matrix with n rows and n columns, its determinant is denoted ⁠\(\det(A)\)⁠, ⁠\(\det A\)⁠, \[\begin{vmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{vmatrix}\] or \[\det\begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{bmatrix}.\]

There are various equivalent ways to define the determinant of a square matrix A, 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.

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, \[\det (AB)=\det(A)\det(B)\] and \[\begin{vmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \ldots & a_{1,n} \\ 0 & a_{2,2} & a_{2,3} & \ldots & a_{2,n} \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \ldots & 0 & a_{n-1,n-1} & a_{n-1,n} \\ 0 & \ldots & \ldots & 0 & a_{n,n} \end{vmatrix} =\begin{vmatrix} a_{1,1} & 0 & 0 & \ldots & 0 \\ a_{2,1} & a_{2,2} & 0 & \ldots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ a_{n-1,1} & a_{n-1,2} & \ldots & a_{n-1,n-1} & 0\\ a_{n,1} & a_{n,2} & \ldots & a_{n,n-1} & a_{n,n} \end{vmatrix} =a_{1,1}\cdots a_{n,n}.\]

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.

An elementary row transformation is a transformation of a matrix of one of the following types, where ⁠\(R_i\)⁠ denotes the ith row of the matrix:

1. Multiplying one row by a scalar: ⁠\(R_i\to \lambda R_i\)⁠
2. Adding to a row a scalar multiple of another row: ⁠\(R_i\to R_i +\lambda R_j\)⁠
3. Exchanging two rows: ⁠\((R_i, R_j)\to (R_j, R_i)\)⁠

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 i and j may be done by the following succession of transformations, where ⁠\(R_i\)⁠ denotes the current ith row (not necessarily the original one): \[R_i\to R_i+R_j, \qquad R_j\to R_j-R_i, \qquad R_i\to R_i+R_j, \qquad R_j\to -R_j\]

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 thedeterminant; 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 ⁠\(R_i\to \lambda R_i\)⁠ amounts to left-multiply the matrix by the diagonal matrix with all diagonal elements equal to 1 except the ith that equals ⁠\(\lambda\)⁠. An elementary transformation ⁠\(R_i\to R_i +\lambda R_j\)⁠ amounts to left-multiplication of the matrix by a triangular matrix in which all diagonal entries are equal to 1, the entry in the ith row and the ijth column is equal to ⁠\(\lambda\)⁠, 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 ⁠\(\det(A)\,\det{B}=\det (AB)\)⁠, it suffices to express A as a product of elementary matrices and apply associative property of matrix product.

Condensed: the full section is in Wikipedia.

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 \[\left. A=\big(a_{i,j}\big)\right|_{i=1...n}^{j=1...n}\] is \[\det(A)=\sum_{\sigma}\operatorname{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma(i)} ,\] where ⁠\(\sigma\)⁠ runs over all permutations of ⁠\(\{1,\ldots,n\}\)⁠.

When ⁠\(A=[~]\)⁠ is the empty matrix, the usual conventions for the empty set and the empty product give ⁠\(\det [~] =1\)⁠. If ⁠\(n=1\)⁠,one has ⁠\(\det [a_{1,1}] = a_{1,1}\)⁠. Otherwise, the above formula is often expanded as \[\det(A)=\sum_{\sigma}\operatorname{sgn}(\sigma) a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}.\]

The Levi-Civita symbol allows a variant of the Leibniz formula that is useful in some contexts. It is denoted \(\varepsilon_{i_1,\ldots,i_n}\) and defined on the n-tuples \((i_1,\ldots,i_n)\) of integers in \(\{1,\ldots,n\}\) as 0 if two of the integers are equal, and otherwise as the signature of the permutation formef by the n-tuple. With the Levi-Civita symbol, the Leibniz formula becomes

\(\det(A) = \sum_{i_1,i_2,\ldots,i_n} \varepsilon_{i_1\cdots i_n} a_{1,i_1} \!\cdots a_{n,i_n},\)

where the sum is taken over all n-tuples of integers in \(\{1,\ldots,n\}.\)

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.

The determinant can be also defined as the unique alternating multilinear form of the columns (or the rows) that takes the value 1 on the identity matrix.

For making this more precise, it is convenient to regard an \(n \times n\) matrix A as being composed of its \(n\) columns, so denoted as \[A = \big ( a_1, \dots, a_n \big ),\] where the column vector \(a_i\) (for each i) is composed of the entries of the matrix in the i-th column. Being a multilinear form means that if \(A = \big ( a_1, \dots, a_i, \ldots, a_n \big )\) and \(A' = \big ( a_1, \dots, a'_i, \ldots, a_n \big )\) are two matrices that differs on one column only, then, for every scalar ⁠\(r\)⁠, one has\[\det\big ( a_1, \dots, a_i+ra'_i , \ldots, a_n\big ) = \det \big ( a_1, \dots, a_i , \ldots, a_n\big )+ r\det \big ( a_1, \dots, a'_i , \ldots, a_n\big ).\] The fact that the form is alternating means that it takes the value 0 if two columns are equal.

To prove that this definition is equivalent with the preceding ones, it suffices to verify the properties of the definition by elementary transformations (above), which is straightforward.

These rules have several further consequences:

• The determinant is a homogeneous function, i.e., \[\det(cA) = c^n\det(A)\] (for an \(n \times n\) matrix \(A\)).
• Interchanging any pair of columns of a matrix multiplies its determinant by −1: \[|a_1, \dots, a_j, \dots a_i, \dots, a_n| = - |a_1, \dots, a_i, \dots, a_j, \dots, a_n|.\] (Given property 2 above, this is equivalent to property 3, which is why property 3 uses the term alternating.) This formula can be applied iteratively when several columns are swapped. For example \[|a_3, a_1, a_2, a_4 \dots, a_n| = - |a_1, a_3, a_2, a_4, \dots, a_n| = |a_1, a_2, a_3, a_4, \dots, a_n|.\] In other words, any permutation of the columns multiplies the determinant by the sign of the permutation.
• If any column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), the determinant is 0. As a special case, if any column has all zero entries, then the determinant is 0.
• Adding a scalar multiple of one column to another column does not change the value of the determinant. This is a consequence of multilinearity and being alternating: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is alternating.
• If \(A\) is a triangular matrix, i.e. \(a_{ij}=0\), whenever \(i>j\) or, alternatively, whenever i

The determinant of the transpose of \(A\) equals the determinant of A:

\(\det\left(A^\textsf{T}\right) = \det(A)\).

This can be proven by inspecting the Leibniz formula. This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing an n × n matrix as being composed of n rows, the determinant is an n-linear function.

The determinant is a multiplicative map, i.e., for square matrices \(A\) and \(B\) of equal size, the determinant of a matrix product equals the product of their determinants:

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

This key fact can be proven by observing that, for a fixed matrix \(B\), both sides of the equation are alternating and multilinear as a function depending on the columns of \(A\). Moreover, they both take the value \(\det B\) when \(A\) is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.

A matrix \(A\) with entries in a field is invertible precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by

\(\det\left(A^{-1}\right) = \frac{1}{\det(A)} = [\det(A)]^{-1}\).

In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size \(n\) over a field \(K\)) forms a group known as the general linear group \(\operatorname{GL}_n(K)\) (respectively, a subgroup called the special linear group \(\operatorname{SL}_n(K) \subset \operatorname{GL}_n(K)\). 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.

Because the determinant respects multiplication and inverses, it is in fact a group homomorphism from \(\operatorname{GL}_n(K)\) into the multiplicative group \(K^\times\) of nonzero elements of \(K\). This homomorphism is surjective and its kernel is \(\operatorname{SL}_n(K)\) (the matrices with determinant one). Hence, by the first isomorphism theorem, this shows that \(\operatorname{SL}_n(K)\) is a normal subgroup of \(\operatorname{GL}_n(K)\), and that the quotient group \(\operatorname{GL}_n(K)/\operatorname{SL}_n(K)\) is isomorphic to \(K^\times\).

The Cauchy-Binet formula is a generalization of that product formula for rectangular matrices. This formula can also be recast as a multiplicative formula for compound matrices whose entries are the determinants of all quadratic submatrices of a given matrix.

Laplace expansion expresses the determinant of a matrix \(A\) recursively in terms of determinants of smaller matrices, known as its minors. The minor \(M_{i,j}\) is defined to be the determinant of the \((n-1) \times (n-1)\) matrix that results from \(A\) by removing the \(i\)-th row and the \(j\)-th column. The expression \((-1)^{i+j}M_{i,j}\) is known as a cofactor. For every \(i\), one has the equality

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

which is called the Laplace expansion along the ith row. For example, the Laplace expansion along the first row (\(i=1\)) gives the following formula:

\(\begin{vmatrix}a&b&c\\ d&e&f\\ g&h&i\end{vmatrix} = a\begin{vmatrix}e&f\\ h&i\end{vmatrix} - b\begin{vmatrix}d&f\\ g&i\end{vmatrix} + c\begin{vmatrix}d&e\\ g&h\end{vmatrix}\)

Unwinding the determinants of these \(2 \times 2\)-matrices gives back the Leibniz formula mentioned above. Similarly, the Laplace expansion along the \(j\)-th column is the equality

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

Laplace expansion can be used iteratively for computing determinants, but this approach is inefficient for large matrices. However, it is useful for computing the determinants of highly symmetric matrix such as the Vandermonde matrix \[\begin{vmatrix} 1 & 1 & 1 & \cdots & 1 \\ x_1 & x_2 & x_3 & \cdots & x_n \\ x_1^2 & x_2^2 & x_3^2 & \cdots & x_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \cdots & x_n^{n-1} \end{vmatrix} = \prod_{1 \leq i < j \leq n} \left(x_j - x_i\right).\]The n-term Laplace expansion along a row or column can be generalized to write an n x n determinant as a sum of \(\tbinom nk\) terms, each the product of the determinant of a k x k submatrix and the determinant of the complementary (n−k) x (n−k) submatrix.

The adjugate matrix \(\operatorname{adj}(A)\) is the transpose of the matrix of the cofactors, that is,

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

For every matrix, one has

\((\det A) I = A\operatorname{adj}A = (\operatorname{adj}A)\,A.\)

Thus the adjugate matrix can be used for expressing the inverse of a nonsingular matrix:

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