The determinantMathworldPlanetmath is an algebraic operation that transforms a square matrixMathworldPlanetmath M into a scalar. This operationMathworldPlanetmath has many useful and important properties. For example, the determinant is zero if and only the matrix M is singularPlanetmathPlanetmath (no inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath exists). The determinant also has an important geometric interpretationMathworldPlanetmathPlanetmath as the area of a parallelogramMathworldPlanetmath, and more generally as the volume of a higher-dimensional parallelepipedMathworldPlanetmath.

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, two-by-two 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.


Let M be an n×n matrix with entries Mij that are elements of a given field11Most scientific and geometric applications deal with matrices made up of real or complex numbersMathworldPlanetmathPlanetmath. 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=|M11M12M1nM21M22M2nMn1Mn2Mnn|=πSnsgn(π)M1π1M2π2Mnπn. (1)

The index π in the above sum varies over all the permutationsMathworldPlanetmath of {1,,n} (i.e., the elements of the symmetric groupMathworldPlanetmathPlanetmath Sn.) Hence, there are n! terms in the defining sum of the determinant. The symbol sgn(π) denotes the parity of the permutation; it is ±1 according to whether π is an even or odd permutationMathworldPlanetmath. Using the Einstein summation convention one can also express the above definition as

detM=ϵπ1π2πnMπ1Mπ212Mπn,n (2)

where we’ve raised the first index so that Mi=jMij, and where


is known as the Levi-Civita permutation symbol.

By way of example, the determinant of a 2×2 matrix is given by


There are six permutations of the numbers 1,2,3, namely

12+3, 23+1, 31+2, 13-2, 32-1, 21-3;

the overset sign indicates the permutation’s signaturePlanetmathPlanetmathPlanetmath. Accordingly, the 3×3 deterimant is a sum of the following 6 terms:


Remarks and important properties

  1. 1.

    The determinant operation converts matrix multiplicationMathworldPlanetmath into scalar multiplication;


    where A,B are square matrices of the same size.

  2. 2.

    The determinant operation is multi-linear, and anti-symmetric with respect to the matrix’s rows and columns. See the multi-linearity attachment for more details.

  3. 3.

    The determinant of a lower triangular, or an upper triangular matrix is the productPlanetmathPlanetmath of the diagonalMathworldPlanetmath entries, since all the other summands in (1) are zero.

  4. 4.

    Similar matricesMathworldPlanetmath ( have the same determinant. To be more precise, let A and X be square matrices with X invertiblePlanetmathPlanetmath. Then,


    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 polynomialMathworldPlanetmathPlanetmath of an endomorphismPlanetmathPlanetmathPlanetmath is definable without using a basis or a matrix, and it turns out that the determinant and trace are two of its coefficients.

  5. 5.

    The determinant of a matrix A is zero if and only if A is singular; that is, if there exists a non-trivial solution to the homogeneous equation

  6. 6.

    The transposeMathworldPlanetmath operation does not change the determinant:

  7. 7.

    The determinant of a diagonalizable transformationMathworldPlanetmath is equal to the product of its eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath, counted with multiplicities.

  8. 8.

    The determinant is homogeneous of degree n. This means that

    det(kM)=kndetM,kis a scalar.
Title determinant
Canonical name Determinant
Date of creation 2013-03-22 12:33:07
Last modified on 2013-03-22 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 PermanentMathworldPlanetmath
Related topic GeneralizedRuizsIdentity