Khanh at 17:10, 4 November 2021

2021-11-04T17:10:11Z

Khanh: Created page with "The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers). The determinant is req..."

2021-10-03T21:01:40Z

Created page with "The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers). The determinant is req..."

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The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers).
The determinant is required to hold these properties:
*It is linear on the rows of the matrix.
:<math>\det \begin{bmatrix} \ddots & \vdots & \ldots \\ \lambda a_1 + \mu b_1 & \cdots & \lambda a_n + \mu b_n \\ \cdots & \vdots & \ddots \end{bmatrix} = \lambda \det \begin{bmatrix} \ddots & \vdots & \cdots \\ a_1 & \cdots & a_n \\ \cdots & \vdots & \ddots \end{bmatrix} + \mu \det \begin{bmatrix} \ddots & \vdots & \cdots \\ b_1 & \cdots & b_n \\ \cdots & \vdots & \ddots \end{bmatrix}</math>
*If the matrix has two equal rows its determinant is zero.
*The determinant of the identity matrix is 1.

It is possible to prove that <math> \det A = \det A^T </math>, making the definition of the determinant on the rows equal to the one on the columns.

==Properties==
*The determinant is zero if and only if the rows are linearly dependent.
*Changing two rows changes the sign of the determinant:
:<math>\det \begin{bmatrix} \cdots \\ \mbox{row A} \\ \cdots \\ \mbox{row B} \\ \cdots \end{bmatrix} = - \det \begin{bmatrix}\cdots \\ \mbox{row B} \\ \cdots \\ \mbox{row A} \\ \cdots \end{bmatrix} </math>

*The determinant is a ''multiplicative map'' in the sense that
:<math>\det(AB) = \det(A)\det(B) \,</math> for all ''n''-by-''n'' matrices <math>A</math> and <math>B</math>.
This is generalized by the Cauchy-Binet formula to products of non-square matrices.

*It is easy to see that <math>\det(rI_n) = r^n \,</math> and thus
:<math>\det(rA) = \det(rI_n \cdot A) = r^n \det(A) \,</math> for all <math>n</math>-by-<math>n</math> matrices <math>A</math> and all scalars <math>r</math>.

*A matrix over a commutative ring ''R'' is invertible if and only if its determinant is a unit in ''R''. In particular, if ''A'' is a matrix over a field such as the real or complex numbers, then ''A'' is invertible if and only if det(''A'') is not zero. In this case we have

:<math>\det(A^{-1}) = \det(A)^{-1}. \,</math>

Expressed differently: the vectors ''v''1,...,''v''''n'' in '''R'''''n'' form a basis if and only if det(''v''1,...,''v''''n'') is non-zero.

A matrix and its transpose have the same determinant:
:<math>\det(A^\top) = \det(A). \,</math>

The determinants of
a complex matrix and of its conjugate transpose
are conjugate:
:<math>\det(A^*) = \det(A)^*. \,</math>

← Older revision		Revision as of 17:10, 4 November 2021
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	are conjugate:		are conjugate:
	:<math>\det(A^) = \det(A)^. \,</math>		:<math>\det(A^) = \det(A)^. \,</math>
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Linear_Algebra/Determinant Determinant, Wikibooks] under a CC BY-SA license

Introduction to Determinants - Revision history

Khanh at 17:10, 4 November 2021

Khanh: Created page with "The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers). The determinant is req..."