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using minors and cofactors Definition 1 Minor of an element a of a determinant is...

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using minors and cofactors Definition 1 Minor of an element a of a determinant is the determinant obtained by deleting its ith row and jth column in which element a lies Minor of an element a IS denoted by M ij Remark Minor of an element of a determinant of order n n 2 is a determinant of order n 1 1 2 3 Example 8 Find the minor of element 6 in the determinant A 4 5 6 8 Solution Since 6 lies in the second row and third column its minor 1 M23 3 8 14 6 obtained by deleting R and C in A is defined by Definition 2 Cofactor of an element denoted A 1 M where M is minor Example 9 Find minors and cofactors of all the elements of the determinant Solution Minor of the element Here a 1 So M Minor of a 3 M 2 Minor of the element a 2 4 M 1 Minor of the element a21 2 M 2 Minor of the element a 1 Now cofactor of a is A So A 1 M 1 3 3 A 2 1 2 M 1 4 4 A 1 1 M 1 2 2 A 2 1 2 M 2 1 1 1 Minor of a M Minor of a M Rationalised 2023 24 Example 10 Find minors and cofactors of the elements a a in the determinant a11 912 a13 a23 a31 a32 a33 Solution By definition of minors and cofactors we have 922 a23 a32 933 a22 a33a23 a32 Cofactor of a A 1 M A22 A33 A23 a32 11 11 11 a12 A 921 922 is given by a13 a a a 3 A32 21 DETERMINANTS 85 ed

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