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Det of a matrix

  • #1
could someone please explain simply how to get the determinate of a 3 * 3 matrix i'm relly stuck i've looked through my text books but it only has examples of how to do it useing a grapgics calculator thanks
 

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  • #2
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Spunky_Dunky said:
could someone please explain simply how to get the determinate of a 3 * 3 matrix i'm relly stuck i've looked through my text books but it only has examples of how to do it useing a grapgics calculator thanks
you break it up into three 2x2 determinents!

http://mathworld.wolfram.com/Determinant.html

look at the first line of eqt. 27
 
  • #3
TD
Homework Helper
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You could do that, or use some properties first to create 0's and then develop to a row or column. There's also a direct way, but it's a bit 'long':

[tex]\begin{gathered}
A = \left( {\begin{array}{*{20}c}
{a_{11} } & {a_{12} } & {a_{13} } \\
{a_{21} } & {a_{22} } & {a_{23} } \\
{a_{31} } & {a_{32} } & {a_{33} } \\
\end{array} } \right) \Rightarrow \det \left( A \right) = \left| {\begin{array}{*{20}c}
{a_{11} } & {a_{12} } & {a_{13} } \\
{a_{21} } & {a_{22} } & {a_{23} } \\
{a_{31} } & {a_{32} } & {a_{33} } \\
\end{array} } \right| \hfill \\ \\
= a_{1,1}\cdot{a}_{2,2}\cdot{a}_{3,3} + a_{1,3}\cdot{a}_{3,2}\cdot{a}_{2,1} + a_{1,2}\cdot{a}_{2,3}\cdot{a}_{3,1} -
a_{1,3}\cdot{a}_{2,2}\cdot{a}_{3,1} - a_{1,1}\cdot{a}_{2,3}\cdot{a}_{3,2} - a_{1,2}\cdot{a}_{2,1}\cdot{a}_{3,3} \hfill \\
\end{gathered} [/tex]
 
  • #4
form Spunky_Dunkey

thanks very much :smile:
 
  • #5
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TD said:
You could do that, or use some properties first to create 0's and then develop to a row or column. There's also a direct way, but it's a bit 'long':

[tex]\begin{gathered}
A = \left( {\begin{array}{*{20}c}
{a_{11} } & {a_{12} } & {a_{13} } \\
{a_{21} } & {a_{22} } & {a_{23} } \\
{a_{31} } & {a_{32} } & {a_{33} } \\
\end{array} } \right) \Rightarrow \det \left( A \right) = \left| {\begin{array}{*{20}c}
{a_{11} } & {a_{12} } & {a_{13} } \\
{a_{21} } & {a_{22} } & {a_{23} } \\
{a_{31} } & {a_{32} } & {a_{33} } \\
\end{array} } \right| \hfill \\ \\
= a_{1,1}\cdot{a}_{2,2}\cdot{a}_{3,3} + a_{1,3}\cdot{a}_{3,2}\cdot{a}_{2,1} + a_{1,2}\cdot{a}_{2,3}\cdot{a}_{3,1} -
a_{1,3}\cdot{a}_{2,2}\cdot{a}_{3,1} - a_{1,1}\cdot{a}_{2,3}\cdot{a}_{3,2} - a_{1,2}\cdot{a}_{2,1}\cdot{a}_{3,3} \hfill \\
\end{gathered} [/tex]


oh, right. that crap.

:tongue:


my calc III prof went over that, mainly as a curiosity. i've used expansion by minors exclusively.


whatever's easiest to you!
 
  • #6
TD
Homework Helper
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I usually expand by minors too, but not before I simplified it first using elementary operations. Having to expand it 'in full' is long too hehe :wink:
 
  • #7
HallsofIvy
Science Advisor
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If you row-reduce the matrix to triangular form, finding the determinant is just multiplying the numbers on the diagonal.
 
  • #8
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TD said:
I usually expand by minors too, but not before I simplified it first using elementary operations. Having to expand it 'in full' is long too hehe :wink:
that's a really good idea. would have really come in handy when i was in 11th grade. :frown: (we had the occasional 4x4 determinant! :yuck: )
 

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