Stranski–Krastanov growth: Difference between revisions

Revision as of 13:00, 30 January 2014

Sarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the determinant of a 3×3 matrix. It is named after the French mathematician Pierre Frédéric Sarrus.

Consider a 3×3 matrix

M={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}},

then its determinant can be computed by the following scheme:

Write out the first 2 columns of the matrix to the right of the 3rd column, so that you have 5 columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed). This yields:

\det(M)={\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}}=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12}.

A similar scheme based on diagonals works for 2x2 matrices:

\det(M)={\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}=a_{11}a_{22}-a_{21}a_{12}.

Both are special cases of the Leibniz formula, which however does not yield similar memorization schemes for larger matrices. Sarrus' rule can also be derived by looking at the Laplace expansion of a 3×3 matrix.

References

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

Sarrus' rule at Planetmath
Linear Algebra: Rule of Sarrus of Determinants at khanacademy.org

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+[[File:Sarrus rule.svg|thumb|right|upright=1.25|''Sarrus' rule'': solid diagonals - dashed diagonals]]
+[[File:Sarrusovo_pravidlo.png|thumb|Rule of Sarrus]]
+'''Sarrus' rule''' or '''Sarrus' scheme''' is a method and a [[mnemonic device|memorization scheme]] to compute the [[determinant]] of a 3&times;3 [[Matrix (mathematics)|matrix]]. It is named after the French mathematician [[Pierre Frédéric Sarrus]].
+Consider a 3&times;3 matrix
+:<math>M=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\  a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}, </math>
+then its determinant can be computed by the following scheme:
+Write out the first 2 columns of the matrix to the right of the 3rd column, so that you have 5 columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed). This yields:
+:<math>\det(M) =\begin{vmatrix} a_{11} & a_{12} & a_{13} \\  a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12}.</math>
+A similar scheme based on diagonals works for 2x2 matrices:
+:<math>\det(M)=\begin{vmatrix} a_{11} & a_{12}  \\  a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{21}a_{12}. </math>
+Both are special cases of the [[Leibniz formula for determinants|Leibniz formula]], which however does not yield similar memorization schemes for larger matrices. Sarrus' rule can also be derived by looking at the [[Laplace expansion]] of a 3&times;3 matrix.
+==References==
+*{{cite book |title=The Pearson Guide to Complete Mathematics for AIEEE |last=Khattar |first=Dinesh |authorlink= |coauthors= |edition=3rd |year=2010 |publisher=Pearson Education India |location= |isbn=978-81-317-2126-1 |page=6-2 |url=http://books.google.de/books?id=7cwSfkQYJ_EC&pg=SA6-PA2 }}
+*{{cite book |title=Analytische Geometrie |last=Fischer |first=Gerd |authorlink= |coauthors= |edition=4th |year=1985 |publisher=Vieweg |location=Wiesbaden |isbn=3-528-37235-4 |page=145 |language=German |url= }}
+==External links==
+*[http://planetmath.org/encyclopedia/RuleOfSarrus.html Sarrus' rule at Planetmath]
+*[http://www.youtube.com/watch?v=4xFIi0JF2AM ''Linear Algebra: Rule of Sarrus of Determinants ''] at khanacademy.org
+[[Category:Linear algebra]]

Stranski–Krastanov growth: Difference between revisions

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