(New page: == Determinants ==  If A is a square matrix then the '''determinant function''' is denoted by '''det '''and '''det(A)''' For an instance we have a 2 x 2 matrix denominated A, theref...)
 
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== Determinants ==
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== Determinants ==
  
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 If A is a square matrix then the '''determinant function''' is denoted by '''det '''and '''det(A)'''
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 If A is a square matrix then the '''determinant function''' is denoted by '''det '''and '''det(A)'''  
  
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For an instance we have a 2 x 2 matrix denominated A, therefore:
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For an instance we have a 2 x 2 matrix denominated A, therefore:  
  
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<br>
  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;'''det(A)''' = [a<sub>11</sub> , &nbsp;a<sub>12</sub>&nbsp;; a<sub>21</sub> , a<sub>22&nbsp;</sub>]
  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;'''det(A)''' = [<span class="Apple-style-span" style="font-size: 11px;">a<sub>11</sub> , &nbsp;a<sub>12</sub> ; a<sub>21</sub> , a<sub>22&nbsp;</sub></span>]
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As we already defined the determinant function we can write some formulas. The formulas for any 2 x 2 and 3 x 3 matrix will be:  
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As we already defined the determinant function we can write some formulas. The formulas for any 2 x 2 and 3 x 3 matrix will be:
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  
 
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; The determinant function for a 2 x 2 matrix is:  
 
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; The determinant function for a 2 x 2 matrix is:  
  
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<br>
  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;'''det(A)''' = [a<sub>11</sub> , a<sub>12</sub>&nbsp;; a<sub>21</sub> , a<sub>22</sub>]
  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;'''det(A)''' = [a<sub>11</sub> , a<sub>12</sub> ; a<sub>21</sub> , a<sub>22</sub>]
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; = '''a<sub>11</sub> * a<sub>22</sub> - a<sub>12</sub> * a'''<sub>'''21'''</sub>  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; = '''a<sub>11</sub> * a<sub>22</sub> - a<sub>12</sub> * a'''<sub>'''21'''</sub>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; The determinant function for a 3 x 3 matrix is:&nbsp;
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  
  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; The determinant function for a 3 x 3 matrix is:&nbsp;
  
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<br>
  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; det(A) = [a<sub>11</sub> , a<sub>12</sub>, a<sub>13</sub> ; a<sub>21</sub> , a<sub>22</sub> , a<sub>23</sub> ; a<sub>31</sub> , a<sub>32</sub> , a<sub>33</sub>]
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; det(A) = [a<sub>11</sub> , a<sub>12</sub>, a<sub>13</sub>&nbsp;; a<sub>21</sub> , a<sub>22</sub> , a<sub>23</sub>&nbsp;; a<sub>31</sub> , a<sub>32</sub> , a<sub>33</sub>]  
  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;= '''(a<sub>11</sub> * a<sub>22</sub> * a<sub>33</sub>) + (a<sub>12</sub> * a<sub>23</sub> * a<sub>31</sub>) + (a<sub>13</sub> * a<sub>21</sub> * a<sub>32</sub>) - (a<sub>12</sub> * a<sub>21</sub> * a<sub>33</sub>) - (a<sub>11</sub> * a<sub>23</sub> * a<sub>32</sub>) - (a<sub>13</sub> * a<sub>22</sub> * a<sub>31</sub>)&nbsp;'''
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;= '''(a<sub>11</sub> * a<sub>22</sub> * a<sub>33</sub>) + (a<sub>12</sub> * a<sub>23</sub> * a<sub>31</sub>) + (a<sub>13</sub> * a<sub>21</sub> * a<sub>32</sub>) - (a<sub>12</sub> * a<sub>21</sub> * a<sub>33</sub>) - (a<sub>11</sub> * a<sub>23</sub> * a<sub>32</sub>) - (a<sub>13</sub> * a<sub>22</sub> * a<sub>31</sub>)&nbsp;'''  
  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;  
  
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<span class="Apple-style-span" style="font-size: 11px;" />
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&lt;math&gt;\left(\begin{array}{cccc}1&amp;2&amp;3&amp;4\\5&amp;6&amp;7&amp;8\end{array}\right)&lt;/math&gt; <br>

Revision as of 14:46, 7 December 2011

Determinants

 If A is a square matrix then the determinant function is denoted by det and det(A)

For an instance we have a 2 x 2 matrix denominated A, therefore:


                                                                                         det(A) = [a11 ,  a12 ; a21 , a22 ]

As we already defined the determinant function we can write some formulas. The formulas for any 2 x 2 and 3 x 3 matrix will be:

                     

                      The determinant function for a 2 x 2 matrix is:


                                                                                         det(A) = [a11 , a12 ; a21 , a22]

                                                                                                = a11 * a22 - a12 * a21

                       

                      The determinant function for a 3 x 3 matrix is: 


                                                                             det(A) = [a11 , a12, a13 ; a21 , a22 , a23 ; a31 , a32 , a33]

                                 = (a11 * a22 * a33) + (a12 * a23 * a31) + (a13 * a21 * a32) - (a12 * a21 * a33) - (a11 * a23 * a32) - (a13 * a22 * a31) 

                                                                                                

<math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math>

Retrieved from "https://www.projectrhea.org/rhea/index.php?title=Determinants_MA265F11Walther&oldid=49044"
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Basic linear algebra uncovers and clarifies very important geometry and algebra.

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