What is Determinant: Definition and 504 Discussions
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible, and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one).
The determinant of a matrix A is denoted det(A), det A, or |A|.
In the case of a 2 × 2 matrix the determinant can be defined as
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{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.\end{aligned}}}
Similarly, for a 3 × 3 matrix A, its determinant is
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{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\[3pt]&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}}
Each determinant of a 2 × 2 matrix in this equation is called a minor of the matrix A. This procedure can be extended to give a recursive definition for the determinant of an n × n matrix, known as Laplace expansion.
Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a matrix, whose roots are the eigenvalues. In geometry, the signed n-dimensional volume of a n-dimensional parallelepiped is expressed by a determinant. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.
J Nutr. 2004 Jul;134(7):http://www.nutrition.org/cgi/content/abstract/134/7/1786.
Ethnicity and race influence the folate status response to controlled folate intakes in young women.
Perry CA, Renna SA, Khitun E, Ortiz M, Moriarty DJ, Caudill MA.
Department of Human Nutrition and Food...
Is one way of looking of the determinant is its the area of the parallelogram formed by the vectors in 2 dimensions, the volume of the parallelpided in 3 dimensions etc. The sign of the determinant tells you something about the relative position of the vectors. This would make the...
Hi I'm new here and I hope that you will be able to give me a lot of help. My english is far to be perfect but sufficiant to asks you a lot of questions... (i hope so :wink:).
First question :
I'm looking for a complete proof (with all steps) of :
\partial _h g = gg_{ij} \partial _h...
I seem to be still having problems with solving the determinant of a 4x4 matrix correctly. There are two methods I can use: Factor/Cofactor and reduction to triangular form.
Ex:
[ 1 2 3 4
-5 2 1 0
6 4 3 2
1 1 1 1]
determinant is 2 based on the calculator
Factor/Cofactor: I...