What is 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.

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Some articles on determinant:

Capelli's Identity - Generalizations - The Mukhin–Tarasov–Varchenko Identity and The Gaudin Model - Statement
... 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
... 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 ...
Medical Priority Dispatch System - Dispatch Determinants
... Each dispatch determinant is made up of three pieces of information, which builds the determinant in a Number-Letter-Number format ... component, a letter A through E (including the Greek character Ω), is the response determinant indicating the potential severity of injury or illness based on information provided by the ... The third and final component, a number, is the sub-determinant and provides more specific information about the patient's specific condition ...
Determinant - Applications - Circulants
... Second order Third order where ω and ω2 are the complex cube roots of 1 ... In general, the nth-order circulant determinant is where ωj is an nth root of 1 ...

More definitions of "determinant":

  • (noun): A square matrix used to solve simultaneous equations.