Determinant

 ( Matrix Theory ) 

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse.

The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries.

The determinant of a matrix is

$\backslash begin\{vmatrix\}$

The determinant of an matrix can be defined in several equivalent ways, the most common being Leibniz formula, which expresses the determinant as a sum of $n!$ (the factorial of ) signed products of matrix entries. It can be computed by the Laplace expansion, which expresses the determinant as a linear combination of determinants of submatrices, or with Gaussian elimination, which allows computing a row echelon form with the same determinant, equal to the product of the diagonal entries of the row echelon form.

Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the matrices that has the four following properties:

1. The determinant of the identity matrix is .
2. The exchange of two rows multiplies the determinant by .
3. Multiplying a row by a number multiplies the determinant by this number.
4. Adding a multiple of one row to another row does not change the determinant.

The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns.

The determinant is invariant under matrix similarity. This implies that, given a linear endomorphism of a finite-dimensional vector space, the determinant of the matrix that represents it on a basis does not depend on the chosen basis. This allows defining the determinant of a linear endomorphism, which does not depend on the choice of a coordinate system.

Determinants occur throughout mathematics. For example, a matrix is often used to represent the in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the . In geometry, the signed -dimensional volume of a -dimensional parallelepiped is expressed by a determinant, and the determinant of a linear endomorphism determines how the orientability and the -dimensional volume are transformed under the endomorphism. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.

The absolute value of is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by .

The absolute value of the determinant together with the sign becomes the signed area of the parallelogram. The signed 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).

To show that is the signed area, one may consider a matrix containing two vectors and representing the parallelogram's sides. The signed area can be expressed as for the angle θ between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the sine this already is the signed area, yet it may be expressed more conveniently using the cosine of the complementary angle to a perpendicular vector, e.g. , so that becomes the signed area in question, which can be determined by the pattern of the scalar product to be equal to according to the following equations:

$\backslash text\{Signed\; area\}\; =$

 |\boldsymbol{u}|\,|\boldsymbol{v}|\,\sin\,\theta = \left|\boldsymbol{u}^\perp\right|\,\left|\boldsymbol{v}\right|\,\cos\,\theta' =
 \begin{pmatrix} -c \\ a \end{pmatrix} \cdot \begin{pmatrix} b \\ d \end{pmatrix} = ad - bc.
     

Thus the determinant gives the area scale 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 preserves area and orientation.

If an Real number matrix A is written in terms of its column vectors $A\; =\; \backslash left\backslash begin\{array\}\{c|c|c|c\}$, then

 A\begin{pmatrix}1 \\ 0\\ \vdots \\0\end{pmatrix} = \mathbf{a}_1, \quad
 A\begin{pmatrix}0 \\ 1\\ \vdots \\0\end{pmatrix} = \mathbf{a}_2, \quad
                                                          \ldots, \quad
 A\begin{pmatrix}0 \\0 \\ \vdots \\1\end{pmatrix} = \mathbf{a}_n.
     

This means that $A$ maps the unit Hypercube to the n-dimensional parallelotope defined by the vectors $\backslash mathbf\{a\}\_1,\; \backslash mathbf\{a\}\_2,\; \backslash ldots,\; \backslash mathbf\{a\}\_n,$ the region $P\; =\; \backslash left\backslash \{c\_1\; \backslash mathbf\{a\}\_1\; +\; \backslash cdots\; +\; c\_n\backslash mathbf\{a\}\_n\; \backslash mid\; 0\; \backslash leq\; c\_i\backslash leq\; 1\; \backslash \; \backslash forall\; i\backslash right\backslash \}$ ($\backslash forall$ stands for "for all" as a logical symbol.)

The determinant gives the signed n-dimensional volume of this parallelotope, $\backslash det(A)\; =\; \backslash pm\; \backslash text\{vol\}(P),$ and hence describes more generally the n-dimensional volume scale factor of the linear transformation produced by A. (The sign shows whether the transformation preserves or reverses orientation.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully n-dimensional, which indicates that the dimension of the image of A is less than n. This means that A produces a linear transformation which is neither onto nor one-to-one, and so is not invertible.

$A\; =\; \backslash begin\{bmatrix\}$

 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}
     

The entries $a\_\{1,1\}$ etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are in a commutative ring.

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

$\backslash begin\{vmatrix\}$

 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}
     

There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.

= aei + bfg + cdh - ceg - bdi - afh.\ 
     

Given a matrix

$A=\backslash begin\{bmatrix\}$

$\backslash det(A)=\backslash begin\{vmatrix\}$

Using pi notation for the product, this can be shortened into

$\backslash det(A)\; =\; \backslash sum\_\{\backslash sigma\; \backslash in\; S\_n\}\; \backslash left(\; \backslash sgn(\backslash sigma)\; \backslash prod\_\{i=1\}^n\; a\_\{i,\backslash sigma(i)\}\backslash right)$.

The Levi-Civita symbol $\backslash varepsilon\_\{i\_1,\backslash ldots,i\_n\}$ is defined on the - of integers in $\backslash \{1,\backslash ldots,n\backslash \}$ as if two of the integers are equal, and otherwise as the signature of the permutation defined by the n-tuple of integers. With the Levi-Civita symbol, the Leibniz formula becomes

$\backslash det(A)\; =\; \backslash sum\_\{i\_1,i\_2,\backslash ldots,i\_n\}\; \backslash varepsilon\_\{i\_1\backslash cdots\; i\_n\}\; a\_\{1,i\_1\}\; \backslash !\backslash cdots\; a\_\{n,i\_n\},$

$A\; =\; \backslash big\; (\; a\_1,\; \backslash dots,\; a\_n\; \backslash big\; ),$

2. $\backslash det\backslash left(I\backslash right)\; =\; 1$, where $I$ is an identity matrix.
4. The determinant is multilinear map: if the jth column of a matrix $A$ is written as a linear combination $a\_j\; =\; r\; \backslash cdot\; v\; +\; w$ of two v and w and a number r, then the determinant of A is expressible as a similar linear combination:
5. : $\backslash begin\{align\}|A|$

 &= \big | a_1, \dots, a_{j-1}, r \cdot v + w, a_{j+1}, \dots, a_n | \\
 &= r \cdot | a_1, \dots, v, \dots a_n | + | a_1, \dots, w, \dots, a_n |
     

2. The determinant is alternating form: whenever two columns of a matrix are identical, its determinant is 0:
3. : $|\; a\_1,\; \backslash dots,\; v,\; \backslash dots,\; v,\; \backslash dots,\; a\_n|\; =\; 0.$

If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any $n\; \backslash times\; n$ matrix A a number that satisfies these three properties.Serge Lang, Linear Algebra, 2nd Edition, Addison-Wesley, 1971, pp 173, 191. This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula.

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 (if the columns are linearly dependent, by property 3) or else ±1 (by property 1 and 3 - the minus sign appears when the columns are permuted according to an odd permutation), 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.

• The determinant is a homogeneous function, i.e., $$\backslash det(cA)\; =\; c^n\backslash det(A)$$ (for an $n\; \backslash times\; n$ matrix $A$).
• Interchanging any pair of columns of a matrix multiplies its determinant by −1. This follows from the determinant being multilinear and alternating (properties 2 and 3 above): $$|a\_1,\; \backslash dots,\; a\_j,\; \backslash dots\; a\_i,\; \backslash dots,\; a\_n|\; =\; -\; |a\_1,\; \backslash dots,\; a\_i,\; \backslash dots,\; a\_j,\; \backslash dots,\; a\_n|.$$ This formula can be applied iteratively when several columns are swapped. For example $$|a\_3,\; a\_1,\; a\_2,\; a\_4\; \backslash dots,\; a\_n|\; =\; -\; |a\_1,\; a\_3,\; a\_2,\; a\_4,\; \backslash dots,\; a\_n|\; =\; |a\_1,\; a\_2,\; a\_3,\; a\_4,\; \backslash dots,\; a\_n|.$$ Yet more generally, any permutation of the columns multiplies the determinant by the sign of the permutation.
• If some column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), the determinant is 0. As a special case, this includes: if some column is such that all its entries are zero, then the determinant of that matrix 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 alternative: 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 diagonal matrix (without changing the determinant). For such a matrix, using the linearity in each column reduces to the identity matrix, in which case the stated formula holds by the very first characterizing property of determinants. Alternatively, this formula can also be deduced from the Leibniz formula, since the only permutation $\backslash sigma$ which gives a non-zero contribution is the identity permutation.

$A\; =\; \backslash begin\{bmatrix\}$

-2 & -1 & 2 \\
 2 & 1 & 4 \\
-3 & 3 & -1
     

 -3 & -1 & 2 \\
 3 & 1 & 4 \\
 0 & 3 & -1
     

 -3 & 5 & 2 \\
 3 & 13 & 4 \\
 0 & 0 & -1
     

 5 & -3 & 2 \\
13 & 3 & 4 \\
 0 & 0 & -1
     

18 & -3 & 2 \\
 0 & 3 & 4 \\
 0 & 0 & -1
     

Combining these equalities gives $|A|\; =\; -|E|\; =\; -(18\; \backslash cdot\; 3\; \backslash cdot\; (-1))\; =\; 54.$

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

$\backslash det(AB)\; =\; \backslash det\; (A)\; \backslash det\; (B)$

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 $\backslash det\; B$ when $A$ is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.Alternatively, proves this result using the functoriality of the exterior power.

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

$\backslash det\backslash left(A^\{-1\}\backslash right)\; =\; \backslash frac\{1\}\{\backslash det(A)\}\; =\; \backslash det(A)^\{-1\}$.

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 $\backslash operatorname\{GL\}\_n(K)$ (respectively, a subgroup called the special linear group $\backslash operatorname\{SL\}\_n(K)\; \backslash subset\; \backslash operatorname\{GL\}\_n(K)$. 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 matrix), and the special unitary group.

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

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 matrix whose entries are the determinants of all quadratic submatrices of a given matrix.

$\backslash det(A)\; =\; \backslash sum\_\{j=1\}^n\; (-1)^\{i+j\}\; a\_\{i,j\}\; M\_\{i,j\},$

 \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}
     

$\backslash det(A)=\; \backslash sum\_\{i=1\}^n\; (-1)^\{i+j\}\; a\_\{i,j\}\; M\_\{i,j\}.$

           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).
     

$(\backslash operatorname\{adj\}(A))\_\{i,j\}\; =\; (-1)^\{i+j\}\; M\_\{ji\}.$

For every matrix, one has.

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

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

$A^\{-1\}\; =\; \backslash frac\; 1\{\backslash det\; A\}\backslash operatorname\{adj\}A.$

$\backslash begin\{align\}$

A similar result holds when $D$ is invertible, namely

$\backslash begin\{align\}$

If the blocks are square matrices of the same size further formulas hold. For example, if $C$ and $D$ commutativity (i.e., $CD=DC$), then

For $A\; =\; D$ and $B\; =\; C$, the following formula holds (even if $A$ and $B$ do not commute).

$\backslash det\backslash left(I\_\backslash mathit\{m\}\; +\; AB\backslash right)\; =\; \backslash det\backslash left(I\_\backslash mathit\{n\}\; +\; BA\backslash right),$

where I _m and I _n are the and identity matrices, respectively.

From this general result several consequences follow.

A generalization is $\backslash det\backslash left(Z\; +\; AWB\backslash right)\; =\; \backslash det\backslash left(\; Z\backslash right)\; \backslash det\backslash left(W\; \backslash right)\; \backslash det\backslash left(W^\{-1\}\; +\; B\; Z^\{-1\}\; A\backslash right)$(see Matrix determinant lemma), where Z is an invertible matrix and W is an invertible matrix.

However, for positive semidefinite matrices $A$, $B$ and $C$ of equal size, $$\backslash det(A\; +\; B\; +\; C)\; +\; \backslash det(C)\; \backslash geq\; \backslash det(A\; +\; C)\; +\; \backslash det(B\; +\; C)\backslash text\{,\}$$ with the corollary $$\backslash det(A\; +\; B)\; \backslash geq\; \backslash det(A)\; +\; \backslash det(B)\backslash text\{.\}$$

Brunn–Minkowski theorem implies that the th root of determinant is a concave function, when restricted to Hermitian matrix positive-definite $n\backslash times\; n$ matrices. Therefore, if and are Hermitian positive-definite $n\backslash times\; n$ matrices, one has $$\backslash sqrtn\{\backslash det(A+B)\}\backslash geq\backslash sqrtn\{\backslash det(A)\}+\backslash sqrtn\{\backslash det(B)\},$$ since the th root of the determinant is a homogeneous function.

$\backslash det(A+B)\; =\; \backslash det(A)\; +\; \backslash det(B)\; +\; \backslash text\{tr\}(A)\backslash text\{tr\}(B)\; -\; \backslash text\{tr\}(AB).$

$\backslash det(A)\; =\; \backslash prod\_\{i=1\}^n\; \backslash lambda\_i=\backslash lambda\_1\backslash lambda\_2\backslash cdots\backslash lambda\_n.$

From this, one immediately sees that the determinant of a matrix $A$ is zero if and only if $0$ is an eigenvalue of $A$. In other words, $A$ is invertible if and only if $0$ is not an eigenvalue of $A$.

The characteristic polynomial is defined as,

$\backslash chi\_A(t)\; =\; \backslash det(t\; \backslash cdot\; I\; -\; A).$

$\backslash chi\_A(\backslash lambda)\; =\; 0.$

A 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

$A\_k\; :=\; \backslash begin\{bmatrix\}$

 a_{1,1} & a_{1,2} & \cdots & a_{1,k} \\
 a_{2,1} & a_{2,2} & \cdots & a_{2,k} \\
  \vdots &  \vdots & \ddots &  \vdots \\
 a_{k,1} & a_{k,2} & \cdots & a_{k,k}
     

being positive, for all $k$ between $1$ and $n$.

$\backslash det(\backslash exp(A))\; =\; \backslash exp(\backslash operatorname\{tr\}(A))$

or, for real matrices ,

$\backslash operatorname\{tr\}(A)\; =\; \backslash log(\backslash det(\backslash exp(A))).$

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

$\backslash exp(L)\; =\; A$

the determinant of is given by

$\backslash det(A)\; =\; \backslash exp(\backslash operatorname\{tr\}(L)).$

For example, for , , and , respectively,

$\backslash begin\{align\}$

 \det(A) &= \frac{1}{2}\left(\left(\operatorname{tr}(A)\right)^2 -  \operatorname{tr}\left(A^2\right)\right), \\
 \det(A) &= \frac{1}{6}\left(\left(\operatorname{tr}(A)\right)^3 - 3\operatorname{tr}(A) ~ \operatorname{tr}\left(A^2\right) + 2 \operatorname{tr}\left(A^3\right)\right), \\
 \det(A) &= \frac{1}{24}\left(\left(\operatorname{tr}(A)\right)^4 - 6\operatorname{tr}\left(A^2\right)\left(\operatorname{tr}(A)\right)^2 + 3\left(\operatorname{tr}\left(A^2\right)\right)^2 + 8\operatorname{tr}\left(A^3\right)~\operatorname{tr}(A) - 6\operatorname{tr}\left(A^4\right)\right).
     

cf. Cayley-Hamilton theorem. Such expressions are deducible from combinatorial arguments, Newton's identities, or the Faddeev–LeVerrier algorithm. That is, for generic , the signed constant term of the characteristic polynomial, determined recursively from

$c\_n\; =\; 1;\; ~~~c\_\{n-m\}\; =\; -\backslash frac\{1\}\{m\}\backslash sum\_\{k=1\}^m\; c\_\{n-m+k\}\; \backslash operatorname\{tr\}\backslash left(A^k\backslash right)\; ~~(1\; \backslash le\; m\; \backslash le\; n)~.$

In the general case, this may also be obtained fromA proof can be found in the Appendix B of

$\backslash det(A)\; =\; \backslash sum\_\{\backslash begin\{array\}\{c\}k\_1,k\_2,\backslash ldots,k\_n\; \backslash geq\; 0\backslash \backslash k\_1+2k\_2+\backslash cdots+nk\_n=n\backslash end\{array\}\}\backslash prod\_\{l=1\}^n\; \backslash frac\{(-1)^\{k\_l+1\}\}\{l^\{k\_l\}k\_l!\}\; \backslash operatorname\{tr\}\backslash left(A^l\backslash right)^\{k\_l\},$

where the sum is taken over the set of all integers satisfying the equation

$\backslash sum\_\{l=1\}^n\; lk\_l\; =\; n.$

The formula can be expressed in terms of the complete exponential Bell polynomial of n arguments s _l = −( l – 1)! tr( A ^l) as

$\backslash det(A)\; =\; \backslash frac\{(-1)^n\}\{n!\}\; B\_n(s\_1,\; s\_2,\; \backslash ldots,\; s\_n).$

This formula can also be used to find the determinant of a matrix with multidimensional indices and . The product and trace of such matrices are defined in a natural way as

$(AB)^I\_J\; =\; \backslash sum\_K\; A^I\_K\; B^K\_J,\; \backslash operatorname\{tr\}(A)\; =\; \backslash sum\_I\; A^I\_I.$

An important arbitrary dimension identity can be obtained from the Mercator series expansion of the logarithm when the expansion converges. If every eigenvalue of A is less than 1 in absolute value,

$\backslash det(I\; +\; A)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{1\}\{k!\}\; \backslash left(-\backslash sum\_\{j=1\}^\backslash infty\; \backslash frac\{(-1)^j\}\{j\}\; \backslash operatorname\{tr\}\backslash left(A^j\backslash right)\backslash right)^k\backslash ,,$

where is the identity matrix. More generally, if

$\backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{1\}\{k!\}\; \backslash left(-\backslash sum\_\{j=1\}^\backslash infty\; \backslash frac\{(-1)^j\; s^j\}\{j\}\backslash operatorname\{tr\}\backslash left(A^j\backslash right)\backslash right)^k\backslash ,,$

is expanded as a formal power series in then all coefficients of for are zero and the remaining polynomial is .

$\backslash operatorname\{tr\}\backslash left(I\; -\; A^\{-1\}\backslash right)\; \backslash le\; \backslash log\backslash det(A)\; \backslash le\; \backslash operatorname\{tr\}(A\; -\; I)$

with equality if and only if . This relationship can be derived via the formula for the Kullback–Leibler divergence between two multivariate normal distributions.

Also,

$\backslash frac\{n\}\{\backslash operatorname\{tr\}\backslash left(A^\{-1\}\backslash right)\}\; \backslash leq\; \backslash det(A)^\backslash frac\{1\}\{n\}\; \backslash leq\; \backslash frac\{1\}\{n\}\backslash operatorname\{tr\}(A)\; \backslash leq\; \backslash sqrt\{\backslash frac\{1\}\{n\}\backslash operatorname\{tr\}\backslash left(A^2\backslash right)\}.$

These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that the harmonic mean is less than the geometric mean, which is less than the arithmetic mean, which is, in turn, less than the root mean square.

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

where $\backslash operatorname\{adj\}(A)$ denotes the adjugate of $A$. In particular, if $A$ is invertible, we have

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

Expressed in terms of the entries of $A$, these are

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

Yet another equivalent formulation is

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

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

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

This identity is used in describing associated to certain matrix . For example, the special linear group $\backslash operatorname\{SL\}\_n$ is defined by the equation $\backslash det\; A\; =\; 1$. The above formula shows that its Lie algebra is the special linear Lie algebra $\backslash mathfrak\{sl\}\_n$ consisting of those matrices having trace zero.

Writing a $3\; \backslash times\; 3$ matrix as where $a,\; b,c$ are column vectors of length 3, then the gradient over one of the three vectors may be written as the cross product of the other two:

$\backslash begin\{align\}$

 \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}.
     

Determinants proper originated separately from the work of Seki Takakazu in 1683 in Japan and parallelly of Leibniz in 1693.Cajori, F. A History of Mathematics p. 80A Brief History of Linear Algebra and Matrix Theory at: stated, without proof, Cramer's rule. Both Cramer and also were led to determinants by the question of passing through a given set of points.

Vandermonde (1771) first recognized determinants as independent functions.Campbell, H: "Linear Algebra With Applications", pages 111–112. Appleton Century Crofts, 1971 gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case.Muir, Sir Thomas, The Theory of Determinants in the historical Order of Development London,. Immediately following, Lagrange (1773) treated determinants of the second and third order and applied it 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 "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the discriminant of a quadratic form. 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 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,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).Origins of mathematical terms: http://jeff560.tripod.com/d.html 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.History of matrices and determinants: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html With him begins the theory in its generality.

used the functional determinant which Sylvester later called the Jacobian. In his memoirs in ''Crelle's Journal'' 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|Arthur Cayley]] began their work.  introduced the modern notation for the determinant using vertical bars.History of matrix notation: http://jeff560.tripod.com/matrices.html
     

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, Otto Hesse, and Sylvester; persymmetric determinants by Sylvester and Hermann Hankel; by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and , in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessian matrix by Sylvester; and symmetric gauche determinants by Trudi. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.

$x\_i\; =\; \backslash frac\{\backslash det(A\_i)\}\{\backslash det(A)\}\; \backslash qquad\; i\; =\; 1,\; 2,\; 3,\; \backslash ldots,\; n$

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.

$\backslash det(A\_i)\; =$

 \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

$A\backslash ,\; \backslash operatorname\{adj\}(A)\; =\; \backslash operatorname\{adj\}(A)\backslash ,\; A\; =\; \backslash det(A)\backslash ,\; I\_n.$

Cramer's rule can be implemented in $\backslash operatorname\; O(n^3)$ time, which is comparable to more common methods of solving systems of linear equations, such as LU decomposition, QR decomposition, or singular value decomposition.

$W(f\_1,\; \backslash ldots,\; f\_n)(x)\; =$

 \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)
     

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. Another such use of the determinant is the resultant, which gives a criterion when two have a common root.

$$\backslash mathbf\{a\backslash times\; b\}\; =\; \backslash det$$

 \mathbf{i}&\mathbf{j}&\mathbf{k}\\
  a_1 & a_2 & a_3 \\
  b_1 & b_2 & b_3 \\
     

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

$\backslash frac\{\backslash operatorname\{volume\}(f(S))\}\{\backslash operatorname\{volume\}(S)\}\; =\; \backslash sqrt\{\backslash det\backslash left(A^\backslash textsf\{T\}\; A\backslash right)\}.$

By calculating the volume of the tetrahedron bounded by four points, they can be used to identify . The volume of any tetrahedron, given its vertices $a,\; b,\; c,\; d$, $\backslash frac\; 1\; 6\; \backslash cdot\; |\backslash det(a-b,b-c,c-d)|$, or any other combination of pairs of vertices that 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

$f:\; \backslash mathbf\; R^n\; \backslash rightarrow\; \backslash mathbf\; R^n,$

the Jacobian matrix is the matrix whose entries are given by the partial derivatives

$D(f)\; =\; \backslash left(\backslash frac\; \{\backslash partial\; f\_i\}\{\backslash partial\; x\_j\}\backslash right)\_\{1\; \backslash leq\; i,\; j\; \backslash leq\; n\}.$

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 is given by

$\backslash int\_\{f(U)\}\; \backslash phi(\backslash mathbf\{v\})\backslash ,\; d\backslash mathbf\{v\}\; =\; \backslash int\_U\; \backslash phi(f(\backslash mathbf\{u\}))\; \backslash left|\backslash det(\backslash operatorname\{D\}f)(\backslash mathbf\{u\})\backslash right|\; \backslash ,d\backslash mathbf\{u\}.$

The Jacobian also occurs in the inverse function theorem.

When applied to the field of Cartography, the determinant can be used to measure the rate of expansion of a map near the poles.

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

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

$T\; :\; V\; \backslash to\; V$

A matrix $A\; \backslash in\; \backslash operatorname\{Mat\}\_\{n\; \backslash times\; n\}(R)$ is invertible (in the sense that there is an inverse matrix whose entries are in $R$) if and only if its determinant is an invertible element in $R$. For $R\; =\; \backslash mathbf\; Z$, this means that the determinant is +1 or −1. Such a matrix is called unimodular.

The determinant being multiplicative, it defines a group homomorphism

$\backslash operatorname\{GL\}\_n(R)\; \backslash rightarrow\; R^\backslash times,$

Given a ring homomorphism $f\; :\; R\; \backslash to\; S$, there is a map $\backslash operatorname\{GL\}\_n(f)\; :\; \backslash operatorname\{GL\}\_n(R)\; \backslash to\; \backslash operatorname\{GL\}\_n(S)$ given by replacing all entries in $R$ by their images under $f$. The determinant respects these maps, i.e., the identity

$f(\backslash det((a\_\{i,j\})))\; =\; \backslash det\; ((f(a\_\{i,j\})))$

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 language of category theory, the determinant is a natural transformation between the two functors $\backslash operatorname\{GL\}\_n$ and $(-)^\backslash times$.. See also . Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of , from the general linear group to the multiplicative group,

$\backslash det:\; \backslash operatorname\{GL\}\_n\; \backslash to\; \backslash mathbb\; G\_m.$

$\backslash begin\{align\}$

           \bigwedge^n T: \bigwedge^n V &\rightarrow \bigwedge^n V \\
 v_1 \wedge v_2 \wedge \dots \wedge v_n &\mapsto T v_1 \wedge T v_2 \wedge \dots \wedge T v_n.
     

As $\backslash bigwedge^n\; V$ is one-dimensional, the map $\backslash bigwedge^n\; T$ is given by multiplying with some scalar, i.e., an element in $R$. Some authors such as use this fact to define the determinant to be the element in $R$ satisfying the following identity (for all $v\_i\; \backslash in\; V$):

$\backslash left(\backslash bigwedge^n\; T\backslash right)\backslash left(v\_1\; \backslash wedge\; \backslash dots\; \backslash wedge\; v\_n\backslash right)\; =\; \backslash det(T)\; \backslash cdot\; v\_1\; \backslash wedge\; \backslash dots\; \backslash wedge\; v\_n.$

This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on $n$-tuples of vectors in $R^n$. For this reason, the highest non-zero exterior power $\backslash bigwedge^n\; V$ (as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of $V$ and similarly for more involved objects such as or of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms $\backslash bigwedge^k\; V$ with .,

$\backslash int\; \backslash exp\backslash left-\backslash theta^TA\backslash eta\backslash right\; \backslash ,d\backslash theta\backslash ,d\backslash eta\; =\; \backslash det\; A$

This unusual-looking expression can be understood as a notational trick that rewrites the conventional expression for the determinant

$\backslash det\; A\; =\; \backslash sum\_\{\backslash sigma\; \backslash in\; S\_n\}\backslash sgn(\backslash sigma)a\_\{1,\backslash sigma(1)\}\backslash cdots\; a\_\{n,\backslash sigma(n)\}.$

This form is popular in physics, where it is often used as a stand-in for the Jacobian determinant. The appeal is that, notationally, the integral takes the form of a path integral, such as in the path integral formulation for quantized Hamiltonian mechanics. An example can be found in the theory of Fadeev–Popov ghosts; although this theory may seem rather abstruse, it's best to keep in mind that the use of the ghost fields is little more than a notational trick to express a Jacobian determinant.

The Pfaffian $\backslash mathrm\{Pf\}\backslash ,A$ of a skew-symmetric matrix $A$ is the square-root of the determinant: that is, $\backslash left(\backslash mathrm\{Pf\}\backslash ,A\backslash right)^2=\backslash det\; A.$ The Berezin integral form for the Pfaffian is even more suggestive; it is

$\backslash int\; \backslash exp\backslash left-\; \backslash ,d\backslash theta\; =\; \backslash mathrm\{Pf\}\backslash ,\; A$

$\backslash det\; :\; A\; \backslash to\; F.$

$\backslash det\; (a\; +\; ib+jc+kd)\; =\; a^2\; +\; b^2\; +\; c^2\; +\; d^2$,

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

$\backslash det(I+A)\; =\; \backslash exp(\backslash operatorname\{tr\}(\backslash log(I+A))).$

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

 &= ab \begin{vmatrix}1&0 \\ 0&1\end{vmatrix}
  =  a \begin{vmatrix}1&0 \\ 0&b\end{vmatrix} \\[5mu]
 &=    \begin{vmatrix}a&0 \\ 0&b\end{vmatrix}
  =  b \begin{vmatrix}a&0 \\ 0&1\end{vmatrix}
  = ba \begin{vmatrix}1&0 \\ 0&1\end{vmatrix}
  = ba,