# Example 22 - Chapter 4 Class 12 Determinants (Term 1)

Last updated at Aug. 17, 2021 by Teachoo

Last updated at Aug. 17, 2021 by Teachoo

Transcript

Example 22 Find minors and cofactors of the elements of the determinant 2 3 5 6 0 4 1 5 7 and verify that a11 A31 + a12 A32 + a13 A33 = 0 First, finding minors and cofactors Minor of a11 = M11 = 2 3 5 6 0 4 1 5 7 = 0 4 5 7 = 0 5(4) = 20 Minor of a12 = M12 = 2 3 5 6 0 4 1 5 7 = 6 4 1 7 = 6(-7) 1(4) = 46 Minor of a13 = M13 = 2 3 5 6 0 4 1 5 7 = 6 0 1 5 = 6(5) 0 = 30 Minor of a21 = M21 = 2 3 5 6 0 4 1 5 7 = 3 5 5 7 = ( 3)( 7) 5(5) = 4 Minor of a22 = M22= 2 3 5 6 0 4 1 5 7 = 2 5 1 7 = 2( 7) 1(5) = 19 Minor of a23 = M23 = 2 3 5 6 0 4 1 5 7 = 2 3 1 5 = 10 1(-3) = 13 Minor of a31 = M31 = 2 3 5 6 0 4 1 5 7 = 3 5 0 4 = 3(4) 0 = 12 Minor of a32= M32 = 2 3 5 6 0 4 1 5 7 = 2 5 6 4 = 2(4) 6(5) = 22 Minor of a33 = M33 = 2 3 5 6 0 4 1 5 7 = 2 3 6 0 = 2(0) 6( 3) = 18 Cofactor of a11 = C11 = ( 1)1+1 M11 = ( 1)2 20 = 1 20 = 20 Cofactor of a12 = A12 = ( 1)1+2 M12 = ( 1)3 . (-46) = ( 1) (-46) = 46 Cofactor of a13 = A13 = ( 1)1+3 M13 = ( 1)4 . 30 = (1) 30 = 30 Cofactor of a21 = A21 = ( 1)2+1 M21 = ( 1)3 . ( 4) = ( 1) ( 4) = 4 Cofactor of a22 = A22 = ( 1)2+2 M22 = ( 1)4 . ( 19) = (1) . ( 19) = 19 Cofactor of a23 = A23 = ( 1)2 + 3 M23= ( 1)5 (13) = ( 1) (13) = 13 Cofactor of a31 = A31= ( 1)3 + 1 M31 = ( 1)4 ( 12) = 1 . ( 12) = 12 Cofactor of a32 = A32 = ( 1)3 + 2 M32 = ( 1)5 . ( 22) = ( 1) ( 22) = 22 Cofactor of a33 = A33 = ( 1)3 + 3 M33 = ( 1)6 . (18) = (1) . (18) = 18 Now for 2 3 5 6 0 4 1 5 7 We need to verify a11 A31 + a12 A32 + a13 A33 = 0 Taking L.H.S Here, a11 = 2 , A31 = 12 a12 = 3 , A32 = 22 a13 = 5 , A33 = 18 Putting values a11 A31 + a12 A32 + a13 A33 = 2( 12) + ( 3) (22) + 5 (18) = 24 66 + 90 = 90 + 90 = 0 Hence Verified

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Chapter 4 Class 12 Determinants (Term 1)

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.