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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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----
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----
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''<u>'''Introduction:'''</u>''<u></u''<u</u>'''''<u></u>'''
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<br>
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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:  
 
For an instance we have a 2 x 2 matrix denominated A, therefore:  
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<br>  
 
<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;<math>det(A)=\left(\begin{array}{cccc}a11&a12\\a21&a22\end{array}\right)</math>&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;<math>det(A)=\left(\begin{array}{cccc}a11&a12\\a21&a22\end{array}\right)</math>&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;= ('''a<sub>11</sub> * a<sub>22)</sub> - (a<sub>12</sub> * a'''<sub>'''21'''</sub><sub>'''&nbsp;'''</sub>) &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; &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><sub>'''&nbsp;'''</sub>) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  
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<br>  
 
<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;<math>det(A)=\left(\begin{array}{cccc}a11&a12&a13\\a21&a22&a23\\a31&a32&a33\end{array}\right)</math>
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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;<math>det(A)=\left(\begin{array}{cccc}a11&a12&a13\\a21&a22&a23\\a31&a32&a33\end{array}\right)</math>  
  
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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;= '''(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;= '''(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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==Properties of Determinants==
 
  
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----
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<u>'''''Properties of Determinants:'''''</u>
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<u>Theorem 1</u>: Let A be an n x n matrix then; '''det(A) = det(A<sup>t</sup>)
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<u></u><u></<u></u><u><u></u><u></u<u></u<u></u><strike></strike><sub></sub><sub></sub><u>Theorem 2:</u> If a matrix B results from matrix A by interchanging two different rows (columns) of A, then; '''det(B) = - det(A)&nbsp;'''
  
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<u>Theorem 3:</u> If two rows (columns) of A are equal, then; '''det(A) = 0'''
  
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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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<u>Theorem 4: </u>If a row (column) of A consists entirely of zeros, then; '''det(A) = 0'''
  
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<br>
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<u>Theorem 5:</u> If B obtained from A by multiplying a row (column) of A by a real number k, then;'''det(B) = ''k''det(A) &nbsp; &nbsp; &nbsp;'''
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<u>Theorem 6:</u> If B = [b<sub>ij</sub>] is obained from A = [a<sub>ij</sub>] by adding to each element of the ''r''th row (column) of A, ''k'' times the corresponding element of the ''s''th row (column), ''r'' not equal ''s'', of A, then; '''det(B) = det(A)'''
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<u></u<u>Theorem 7:</u> If a matrix A = [a<sub>ij</sub>] is upper (lower) triangular, then; det(A) = a<sub>11</sub>*a<sub>12</sub>...a<sub>nn </sub>; tha is, the determinant of a triangular matrix is the product of the element on themain diagonal. &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;<br>

Revision as of 16:23, 7 December 2011

Determinants


Introduction:</u<u


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)=\left(\begin{array}{cccc}a11&a12\\a21&a22\end{array}\right) $ 

                                                                                               = (a11 * a22) - (a12 * a21 )                        

                   

                      The determinant function for a 3 x 3 matrix is: 


                                                                               $ det(A)=\left(\begin{array}{cccc}a11&a12&a13\\a21&a22&a23\\a31&a32&a33\end{array}\right) $

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

Properties of Determinants:


Theorem 1: Let A be an n x n matrix then; det(A) = det(At) </<u><u></u<u></u<u>Theorem 2: If a matrix B results from matrix A by interchanging two different rows (columns) of A, then; det(B) = - det(A) 

Theorem 3: If two rows (columns) of A are equal, then; det(A) = 0

Theorem 4: If a row (column) of A consists entirely of zeros, then; det(A) = 0

Theorem 5: If B obtained from A by multiplying a row (column) of A by a real number k, then;det(B) = kdet(A)       Theorem 6: If B = [bij] is obained from A = [aij] by adding to each element of the rth row (column) of A, k times the corresponding element of the sth row (column), r not equal s, of A, then; det(B) = det(A) </u<u>Theorem 7: If a matrix A = [aij] is upper (lower) triangular, then; det(A) = a11*a12...ann ; tha is, the determinant of a triangular matrix is the product of the element on themain diagonal.                                                       

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Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

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