Page 170 - Artificial Intellegence_v2.0_Class_11
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1   1  − 5     (Swapping the positions of a and d
                                       A − 1  =      
                                            − 4 − 2  6          giving minus sign to b and c)
                                               
                                              1   5  −
                                                   
                                          =    4  4 
                                              1  − 3 
                                                   
                                                  2  2 
              Note, If the determinant of a matrix is zero, then the inverse of that matrix does not exist.
              a)  Inverse of 3x3 Matrix: Let’s understand this using an example.

                                                                 2   − 10
                                                                        
                                               Given a matrix  A =    1  02 Calculate A -1
                                                                         
                                                                  1   13   
              We know that the formula of inverse is:
                               1
                           −1
                          A =   (  AdjA )
                              A                                                       ∴  A =  −  3
                         Let’s calculateA first.              0  2       12     10     Let's calculateAdjoint(A)
                                                                    (
                              2  −1 0                     ⇒ 2  13  − − 1) 13  +0 11        2   − 10
                          A = 1  0  2  Expanding along rrow1 ⇒20 2)  + 3 2)  + 0       A =     1  02   
                                                                     (
                                                              (
                                                                       −
                                                                −
                             1   1  3                       ⇒− +⇒ − 3                       1   13   
                                                              41
              Step 1:   Covering the first row and first column.
                                                    2 − 1 0
                                                             02
                                                                      0
                                                                        2
                                                           
                                                    1  0 2 ⇒  13  ⇒ −⇒ −    2
                                                    1   1 3  
              Step 2:   Covering the first row and second column.
                                                       2 − 1 0
                                                                12
                                                                        32
                                                       1  0 2 ⇒  13  ⇒ −⇒    1
                                                              
                                                       1   1 3  
              Step 3:   Cover the first row and third column.
                                                      2 − 1 0
                                                               10
                                                                        10
                                                      1  0 2 ⇒  11   ⇒ −⇒    1
                                                              
                                                      1   1 3  
              Step 4:   Cover the second row and first column.
                                                     2 − 1 0
                                                              − 10
                                                                         30
                                                     1  0 2 ⇒  1  3 ⇒− −⇒ −    3
                                                            
                                                     1   1 3  
              Step 5:   Cover the second row and second column.
                                                      2 − 1 0
                                                               20
                                                                        60
                                                              
                                                      1  0 2 ⇒  13   ⇒ −⇒    6
                                                      1   1 3  





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