algebra-Determinant

May 28th, 2015
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Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations. As shown by Cramer's rule, a nonhomogeneous system of linear equations has a unique solution iff the determinant of the

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Algebra>Linear Algebra>Determinants>MathWorld Contributors>Budney>Interactive Entries>Interactive Demonstrations>DeterminantDeterminants are mathematical objects that are very useful in the analysis and solution ofsystems of linear equations. As shown byCramer's rule, a nonhomogeneous system of linear equations has a unique solutioniffthe determinant of the system'smatrixisnonzero(i.e., thematrixis nonsingular). For example, eliminating,, andfrom the equations(1)(2)(3)gives the expression(4)which is called the determinant for this system of equation. Determinants are defined only forsquare matrices.If the determinant of amatrixis 0, thematrixis said to besingular, and if the determinant is 1, the matrix is said to beunimodular.The determinant of amatrix,(5)is commonly denoted,, or in component notation as,, or(Muir 1960, p.17). Note that the notationmay be more convenient when indicating the absolute value of a determinant, i.e.,instead of. The determinant is implemented inMathematicaasDet[m].Adeterminant is defined to be(6)Adeterminant can be expanded "byminors" to obtain(7)A general determinant for amatrixhas a value(8)with no implied summation overand where(also denoted) is thecofactorofdefined by(9)andis theminorofmatrixformed by eliminating rowand columnfrom. This process

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