a32a33 Cofactor of a21 A 1 M 1 a 2a33a 3 a Remark Expanding the determinant A in Example 21 along R a2 2 1 1 A 1 a 1 a32 86 911 1 2 a A a 2A 2 a13A 3 where A is cofactor of a 11 12 sum of product of elements of R with their corresponding cofactors Similarly A can be calculated by other five ways of expansion that is along R R C C and C Hence A sum of the produ of elements of any row or column with their corresponding cofactors MATHEMATICS Note If elements of a row or column are multiplied with cofactors of any other row or column then their sum is zero For example A a A 1 a 2 A22 a13 A23 a12 913 a32 6 0 5 Solution We have M M 2 1 1 all a12 a13 a11 a12 a13 a31 932 933 Similarly we can try for other rows and columns M 3 M M 2 M 3 a23 a33 5 16 64 1 II 12 33 943 a33 4 and verify that a A3 a 2 A32 a 3 A33 0 31 2 a 12 Example 11 Find minors and cofactors of the elements of the determinant 2 3 5 5 921 923 a31 a33 1 2 10 4 Rationalised 2023 24 7 21 2 RT 0 since R and R are identical a21 922 1 3 a3 a31 932 9 1 a13 a31 933 30 0 30 a12933 a13a32 we have 4 a 13 14 5 19 10 3 13 1 3 0 20 20 A 1 20 20 42 4 46 A 1 2 46 46 a11 a12 a31 a32 1 2 19 19 A23 1 3 13 13 perance