# 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, while 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|. 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

is written and has the value

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.

Read more about Determinant:  Definition, Properties of The Determinant, Calculation, History

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... is known to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of the definitions ... The permanent is defined similarly to the determinant, as a sum of products of sets of matrix entries that lie in distinct rows and columns ... However, where the determinant weights each of these products with a ±1 sign based on the parity of the set, the permanent weights them all with a +1 sign ...
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... Each dispatch determinant is made up of three pieces of information, which builds the determinant in a Number-Letter-Number format ... E (including the Greek character Ω), is the response determinant indicating the potential severity of injury or illness based on information provided by ... The third and final component, a number, is the sub-determinant and provides more specific information about the patient's specific condition ...
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... which elements are complex numbers Here the first determinant is understood (as always) as column-determinant of a matrix with non-commutative entries ... The determinant on the right is calculated as if all the elements commute, and putting all x and z on the left, while derivations on the right ...
Capelli's Identity - Statement
... The Capelli identity states that the following differential operators, expressed as determinants, are equal Both sides are differential operators ... The determinant on the left has non-commuting entries, and is expanded with all terms preserving their "left to right" order ... Such a determinant is often called a column-determinant, since it can be obtained by the column expansion of the determinant starting from the first column ...