KermMartian at 19:13, 24 February 2016

2016-02-24T19:13:26Z

← Older revision		Revision as of 19:13, 24 February 2016
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	For 2×2 matrices, the determinant is simply		For 2×2 matrices, the determinant is simply

	~~[[TI-BASIC:Math\|Math]]~~		<math>
	\det\left( \begin{bmatrix} a & b\\c & d \end{bmatrix} \right) = \begin{vmatrix} a & b\\c & d \end{vmatrix} = ad-bc		\det\left( \begin{bmatrix} a & b\\c & d \end{bmatrix} \right) = \begin{vmatrix} a & b\\c & d \end{vmatrix} = ad-bc
	~~[[TI-BASIC:~~/math~~\|/math]]~~		</math>

	For larger matrices, the determinant can be computed using the [wikipedia:Laplace_expansion Laplace expansion], which allows you to express the determinant of an n×n matrix in terms of the determinants of (n-1)×(n-1) matrices. However, since the Laplace expansion takes <math>O\left( n! \right)</math> operations, the method usually used in calculators is [http://mathworld.wolfram.com/GaussianElimination.html Gaussian elimination], which only needs <math>O\left( n^3 \right)</math> operations.		For larger matrices, the determinant can be computed using the [wikipedia:Laplace_expansion Laplace expansion], which allows you to express the determinant of an n×n matrix in terms of the determinants of (n-1)×(n-1) matrices. However, since the Laplace expansion takes <math>O\left( n! \right)</math> operations, the method usually used in calculators is [http://mathworld.wolfram.com/GaussianElimination.html Gaussian elimination], which only needs <math>O\left( n^3 \right)</math> operations.
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	The matrix is first decomposed into a unit lower-triangular matrix and an upper-triangular matrix using elementary row operations:		The matrix is first decomposed into a unit lower-triangular matrix and an upper-triangular matrix using elementary row operations:

	~~[[TI-BASIC:Math\|Math]]~~		<math>
	\begin{pmatrix}{1}&{}&{}\\ {\vdots}&{\ddots}&{}\\ {\times}&{\cdots}&{1}\end{pmatrix}		\begin{pmatrix}{1}&{}&{}\\ {\vdots}&{\ddots}&{}\\ {\times}&{\cdots}&{1}\end{pmatrix}
	\begin{pmatrix}{\times}&{\cdots}&{\times}\\ {}&{\ddots}&{\vdots}\\ {}&{}&{\times}\end{pmatrix}		\begin{pmatrix}{\times}&{\cdots}&{\times}\\ {}&{\ddots}&{\vdots}\\ {}&{}&{\times}\end{pmatrix}
	~~[[TI-BASIC:~~/math~~\|/math]]~~		</math>

	The determinant is then calculated as the product of the diagonal elements of the upper-triangular matrix.		The determinant is then calculated as the product of the diagonal elements of the upper-triangular matrix.
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	* [[TI-BASIC:Identity\|Identity(]]		* [[TI-BASIC:Identity\|Identity(]]
	* [[TI-BASIC:Ref\|Ref(]]		* [[TI-BASIC:Ref\|Ref(]]
	* [[TI-BASIC:Rref\|Rref(]][[Category:TI-BASIC]]		* [[TI-BASIC:Rref\|Rref(]]
			[[Category:TI-BASIC]]
	[[Category:TIBD]]		[[Category:TIBD]]

Maintenance script: Initial automated import

2016-02-24T18:04:46Z

Initial automated import

New page

{{Template:TI-BASIC:Command
|picture=DET.GIF
|summary=Calculates the determinant of a square matrix.
|syntax=det(''matrix'')
|location=Press:
# MATRX (83) or 2nd MATRX (83+ or higher) to access the matrix menu
# LEFT to access the MATH submenu
# ENTER to select det(.
|compatibility=TI-83/84/+/SE
|size=1 byte
}}

The det( command calculates the [http://mathworld.wolfram.com/Determinant.html determinant] of a square matrix. If its argument is not a square matrix, [[TI-BASIC:Errors#invaliddim|ERR:INVALID DIM]] will be thrown.

= Advanced Uses =

If [A] is an N×N matrix, then the roots of det([A]-X identity(N)) are the [http://mathworld.wolfram.com/Eigenvalue.html eigenvalues] of [A].

= Formulas =

For 2×2 matrices, the determinant is simply

[[TI-BASIC:Math|Math]]
\det\left( \begin{bmatrix} a & b\\c & d \end{bmatrix} \right) = \begin{vmatrix} a & b\\c & d \end{vmatrix} = ad-bc
[[TI-BASIC:/math|/math]]

For larger matrices, the determinant can be computed using the [wikipedia:Laplace_expansion Laplace expansion], which allows you to express the determinant of an n×n matrix in terms of the determinants of (n-1)×(n-1) matrices. However, since the Laplace expansion takes <math>O\left( n! \right)</math> operations, the method usually used in calculators is [http://mathworld.wolfram.com/GaussianElimination.html Gaussian elimination], which only needs <math>O\left( n^3 \right)</math> operations.

The matrix is first decomposed into a unit lower-triangular matrix and an upper-triangular matrix using elementary row operations:

[[TI-BASIC:Math|Math]]
\begin{pmatrix}{1}&{}&{}\\ {\vdots}&{\ddots}&{}\\ {\times}&{\cdots}&{1}\end{pmatrix}
\begin{pmatrix}{\times}&{\cdots}&{\times}\\ {}&{\ddots}&{\vdots}\\ {}&{}&{\times}\end{pmatrix}
[[TI-BASIC:/math|/math]]

The determinant is then calculated as the product of the diagonal elements of the upper-triangular matrix.

= Error Conditions =

* '''[[TI-BASIC:Errors#invaliddim|ERR:INVALID DIM]]''' is thrown when the matrix is not square.

= Related Commands =

* [[TI-BASIC:Identity|Identity(]]
* [[TI-BASIC:Ref|Ref(]]
* [[TI-BASIC:Rref|Rref(]][[Category:TI-BASIC]]
[[Category:TIBD]]

TI-BASIC:Det - Revision history

KermMartian at 19:13, 24 February 2016

Maintenance script: Initial automated import