determinant
Overview
The determinant^{} is an algebraic operation that transforms a square matrix^{} $M$ into a scalar. This operation^{} has many useful and important properties. For example, the determinant is zero if and only the matrix $M$ is singular^{} (no inverse^{} exists). The determinant also has an important geometric interpretation^{} as the area of a parallelogram^{}, and more generally as the volume of a higherdimensional parallelepiped^{}.
The notion of determinant predates matrices and linear transformations. Originally, the determinant was a number associated to a system of $n$ linear equations in $n$ variables. This number “determined” whether the system possessed a unique solution. In this sense, twobytwo determinants were considered by Cardano at the end of the 16th century and ones of arbitrary size (see the definition below) by Leibniz about 100 years later.
Definition
Let $M$ be an $n\times n$ matrix with entries ${M}_{ij}$ that are elements of a given field^{1}^{1}Most scientific and geometric applications deal with matrices made up of real or complex numbers^{}. However, the determinant of a matrix over any field is well defined sense and has all the properties of the more conventional determinant. Indeed, many properties of the determinant remain valid for matrices with entries in a commutative ring.. The determinant of $M$, or $detM$ for short, is the scalar quantity
$$detM=\left\begin{array}{cccc}\hfill {M}_{11}\hfill & \hfill {M}_{12}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {M}_{1n}\hfill \\ \hfill {M}_{21}\hfill & \hfill {M}_{22}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {M}_{2n}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {M}_{n1}\hfill & \hfill {M}_{n2}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {M}_{nn}\hfill \end{array}\right=\sum _{\pi \in {S}_{n}}\mathrm{sgn}(\pi ){M}_{1{\pi}_{1}}{M}_{2{\pi}_{2}}\mathrm{\cdots}{M}_{n{\pi}_{n}}.$$  (1) 
The index $\pi $ in the above sum varies over all the permutations^{} of $\{1,\mathrm{\dots},n\}$ (i.e., the elements of the symmetric group^{} ${S}_{n}$.) Hence, there are $n!$ terms in the defining sum of the determinant. The symbol $\mathrm{sgn}(\pi )$ denotes the parity of the permutation; it is $\pm 1$ according to whether $\pi $ is an even or odd permutation^{}. Using the Einstein summation convention one can also express the above definition as
$$detM={\u03f5}_{{\pi}_{1}{\pi}_{2}\mathrm{\dots}{\pi}_{n}}{M}^{{\pi}_{1}}{}_{1}M^{{\pi}_{2}}{}_{2}\mathrm{\cdots}{M}^{{\pi}_{n}}{}_{n},$$  (2) 
where we’ve raised the first index so that ${M}^{i}{}_{j}={M}_{ij}$, and where
$${\u03f5}_{{\pi}_{1}\mathrm{\dots}{\pi}_{n}}=\mathrm{sgn}(\pi )$$ 
is known as the LeviCivita permutation symbol.
By way of example, the determinant of a $2\times 2$ matrix is given by
$$\left\begin{array}{cc}\hfill {M}_{11}\hfill & \hfill {M}_{12}\hfill \\ \hfill {M}_{21}\hfill & \hfill {M}_{22}\hfill \end{array}\right={M}_{11}{M}_{22}{M}_{12}{M}_{21},$$ 
There are six permutations of the numbers $1,2,3$, namely
$$1\stackrel{+}{2}3,\mathrm{\hspace{0.33em}2}\stackrel{+}{3}1,\mathrm{\hspace{0.33em}3}\stackrel{+}{1}2,\mathrm{\hspace{0.33em}1}\stackrel{}{3}2,\mathrm{\hspace{0.33em}3}\stackrel{}{2}1,\mathrm{\hspace{0.33em}2}\stackrel{}{1}3;$$ 
the overset sign indicates the permutation’s signature^{}. Accordingly, the $3\times 3$ deterimant is a sum of the following $6$ terms:
$$\left\begin{array}{ccc}\hfill {M}_{11}\hfill & \hfill {M}_{12}\hfill & \hfill {M}_{13}\hfill \\ \hfill {M}_{21}\hfill & \hfill {M}_{22}\hfill & \hfill {M}_{23}\hfill \\ \hfill {M}_{31}\hfill & \hfill {M}_{32}\hfill & \hfill {M}_{33}\hfill \end{array}\right=\begin{array}{cc}& \\ \hfill {M}_{11}{M}_{22}{M}_{33}+{M}_{12}{M}_{23}{M}_{31}+{M}_{13}{M}_{21}{M}_{32}& \\ \hfill {M}_{11}{M}_{23}{M}_{32}{M}_{13}{M}_{22}{M}_{31}{M}_{12}{M}_{21}{M}_{33}& \end{array}$$ 
Remarks and important properties

1.
The determinant operation converts matrix multiplication^{} into scalar multiplication;
$$det(AB)=det(A)det(B),$$ where $A,B$ are square matrices of the same size.

2.
The determinant operation is multilinear, and antisymmetric with respect to the matrix’s rows and columns. See the multilinearity attachment for more details.

3.
The determinant of a lower triangular, or an upper triangular matrix is the product^{} of the diagonal^{} entries, since all the other summands in (1) are zero.

4.
Similar matrices^{} (http://planetmath.org/SimilarMatrix) have the same determinant. To be more precise, let $A$ and $X$ be square matrices with $X$ invertible^{}. Then,
$$det(XA{X}^{1})=det(A).$$ In particular, if we let $X$ be the matrix representing a change of basis, this shows that the determinant is independent of the basis. The same is true of the trace of a matrix. In fact, the whole characteristic polynomial^{} of an endomorphism^{} is definable without using a basis or a matrix, and it turns out that the determinant and trace are two of its coefficients.

5.
The determinant of a matrix $A$ is zero if and only if $A$ is singular; that is, if there exists a nontrivial solution to the homogeneous equation
$$A\mathbf{x}=\mathrm{\U0001d7ce}.$$ 
6.
The transpose^{} operation does not change the determinant:
$$det{A}^{\mathrm{T}}=detA.$$ 
7.
The determinant of a diagonalizable transformation^{} is equal to the product of its eigenvalues^{}, counted with multiplicities.

8.
The determinant is homogeneous of degree $n$. This means that
$$det(kM)={k}^{n}detM,k\text{is a scalar.}$$
Title  determinant 

Canonical name  Determinant 
Date of creation  20130322 12:33:07 
Last modified on  20130322 12:33:07 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  24 
Author  rmilson (146) 
Entry type  Definition 
Classification  msc 15A15 
Related topic  LaplaceExpansion 
Related topic  Permanent^{} 
Related topic  GeneralizedRuizsIdentity 