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3.   Determinant of a 3x3 matrix: Determinant of a matrix of order three can be determined by expressing it in terms
                    of second order determinants. This is known as expansion of a determinant along a row (or a column).
                                                     a    a   a        a   a   a
                                                      11   12  13       11  12  13
                                                 A =  a   a   a     A a    a   a
                                                      21   22  23       21  22  23
                                                     a    a   a        a a  a  a
                                                      31   32  33       31  32  33
                                                            Expanding along Row 1
              Following are the steps to calculate determinant of a matrix:

              Step 1:     Multiply first element a  of Row1 by (–1) (1 + 1)  [(–1)sum of suffixes in a ] and with the second order determinant
                                           11
                                                                                  11
                       obtained by deleting the elements of first row and first column of | A | as a  lies in Row1 and Column1.
                                                                                        11
                                                 a 11  a 12  a 13     a   a
                                                                  +
                                                 a   a   a  ⇒ − (  1) 11 a  22  23
                                                  21  22  23        11  a  a
                                                 a 31  a 32  a 33      32  33
                                                            ⇒ a  (a  × a  – a  × a )
                                                                                  23
                                                                        33
                                                                11
                                                                             32
                                                                    22
                                nd
              Step 2:   Multiply 2  element a  of Row1 by (–1)1 + 2 [(–1) sum of suffixes in a ] and the second-order determinant
                                                                                     12
                                           12
                       obtained by deleting elements of the first row and 2  column of | A | as a  lies in Row1 and Column2.
                                                                     nd
                                                                                        12
                                                 a 11  a 12  a 13
                                                             (
                                                                 +
                                                               1
                                                a    a   a ⇒ − ) 12 a  a 21  a 23
                                                 21  22  23         12 a  a
                                                 a 31  a 32  a 33      31  33
                                                            ⇒ –a  (a  × a  – a  × a )
                                                                12
                                                                         33
                                                                              31
                                                                                  23
                                                                    21
              Step 3:     Multiply third element a  of Row1 by (–1)1 + 3 [(–1) sum of suffixes in a 13 ] and the second-order determinant
                                             13
                       obtained by deleting elements of the first row and third column of  | A | as a  lies in Row1 and Column3.
                                                                                          13
                                               a 11  a 12  a 13        a   a
                                               a 21  a 22  a 23  ⇒ − 1 ) 13 a 13  a 21  a 22
                                                                  +
                                                              (
                                               a 31  a 32  a 33         31  32
                                                             ⇒ –a  (a  × a  – a  × a )
                                                                     21
                                                                 13
                                                                          32
                                                                                   22
                                                                               31
              Now the determinant of A, | A | is written as the sum of all three terms obtained in steps 1, 2 and 3 above:
              So Finally combine all three –
                                         = a (a  × a  – a  × a ) – a  (a  × a  – a  × a )
                                       A    11  22   33  32   23   12  21   33   31   23
                                         + a  (a  × a  – a  × a )
                                            13
                                                               22
                                                          31
                                                21
                                                     32
              For example:
              Calculate the determinant of the following matrix.
                                                                  1 23
                                                             A =      4 05   
                                                                  7 80 
                                                                      
              Solution:
                                                       12 3
                                                   A = 40 5   Expanding alongRow1
                                                       78 0
                                                         05      45      40
                                                      = 1     − 2     + 3
                                                         80      70      78
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