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 Ex 4.4, 2 - Write Minors, Cofactors of determinants - Class 12 - Finding Minors and cofactors

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  1. Chapter 4 Class 12 Determinants
  2. Serial order wise
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Ex 4.4, 2 Write Minors and Cofactors of the elements of following determinants: (i) 1﷮0﷮0﷮0﷮1﷮0﷮0﷮0﷮1﷯﷯ Minor of a11 = M11 = 1﷮0﷮0﷮0﷮1﷮0﷮0﷮0﷮1﷯﷯ = 1﷮0﷮0﷮1﷯﷯ = 1(1) – 0 = 1 Minor of a12 = M12 = 1﷮0﷮0﷮0﷮1﷮0﷮0﷮0﷮1﷯﷯ = 0﷮0﷮0﷮1﷯﷯ = 0 – 0 = 0 Minor of a13 = M13 = 1﷮0﷮0﷮0﷮1﷮0﷮0﷮0﷮1﷯﷯ = 0﷮1﷮0﷮0﷯﷯ = 0 – 0 = 0 Minor of a21 = M21 = 1﷮0﷮0﷮0﷮1﷮0﷮0﷮0﷮1﷯﷯ = 0﷮0﷮0﷮1﷯﷯ = 0 – 0 = 0 Minor of a22 = M22 = 1﷮0﷮0﷮0﷮1﷮0﷮0﷮0﷮1﷯﷯ = 1﷮0﷮0﷮1﷯﷯ = 1 – 0 = 1 Minor of a23 = M23 = 1﷮0﷮0﷮0﷮1﷮0﷮0﷮0﷮1﷯﷯ = 1﷮0﷮0﷮0﷯﷯ = 0 – 0 = 0 Minor of a31 = M31 = 1﷮0﷮0﷮0﷮1﷮0﷮0﷮0﷮1﷯﷯ = 0﷮0﷮1﷮0﷯﷯ = 0 – 0 = 0 Minor of a32 = M32 = 1﷮0﷮0﷮0﷮1﷮0﷮0﷮0﷮1﷯﷯ = 1﷮0﷮0﷮0﷯﷯ = 0 – 0 = 0 Minor of a33 = M33 = 1﷮0﷮0﷮0﷮1﷮0﷮0﷮0﷮1﷯﷯ = 1﷮0﷮0﷮1﷯﷯ = 1 – 0 = 1 Cofactor of a11 = A11 = ( – 1)1+1 M11= ( – 1)2 × 1= 1 × 1 = 1 Cofactor of a12 = A12 = ( – 1)1+2 M12= ( – 1)3 0 = 0 Cofactor of a13 = A13 = ( – 1)1+3 M13 = ( – 1)4 0 = 0 Cofactor of a21 = A21 = ( – 1)2+1 M21 = ( – 1)3 0 = 0 Cofactor of a22 = A22 = ( – 1)2+2 M22 = ( – 1)4 1 = 1 . 1 = 1 Cofactor of a23 = A23 = ( – 1)3 + 1 M23 = ( – 1)4 0 = 0 Cofactor of a31 = A31 = ( – 1)3 + 1 M31 = ( – 1)4 0 = 0 Cofactor of a32 = A32 = ( – 1)3 + 2 M32 = ( – 1)5 0 = 0 Cofactor of a33 = A33 = ( – 1)3 + 3 M33 = ( – 1)6 . 1 = 1 . 1 = 1 Ex 4.4, 2 Write Minors and Cofactors of the elements of following determinants: (ii) 1﷮0﷮4﷮3﷮5﷮−1﷮0﷮1﷮2﷯﷯ Minor of a11 = M11= 1﷮0﷮4﷮3﷮5﷮−1﷮0﷮1﷮2﷯﷯= 5﷮−1﷮1﷮2﷯﷯= 5(2) – 1(−1)= 10 + 1= 11 Minor of a12 = M12 = 1﷮0﷮4﷮3﷮5﷮−1﷮0﷮1﷮2﷯﷯ = 3﷮−1﷮0﷮2﷯﷯ = 6 – 0 = 6 Minor of a13 = M13 = 1﷮0﷮4﷮3﷮5﷮−1﷮0﷮1﷮2﷯﷯= 3﷮5﷮0﷮1﷯﷯ = 3 – 0 = 3 Minor of a21 = M21 = 1﷮0﷮4﷮3﷮5﷮−1﷮0﷮1﷮2﷯﷯ = 0﷮4﷮1﷮2﷯﷯ = 0 – 4 = – 4 Minor of a22 = M22 = 1﷮0﷮4﷮3﷮5﷮−1﷮0﷮1﷮2﷯﷯ = 1﷮4﷮0﷮2﷯﷯ = 2 – 0 = 2 Minor of a23 = M23 = 1﷮0﷮4﷮3﷮5﷮−1﷮0﷮1﷮2﷯﷯ = 1﷮0﷮0﷮1﷯﷯ = 1 – 0 = 1 Minor of a31 = M31 = 1﷮0﷮4﷮3﷮5﷮−1﷮0﷮1﷮2﷯﷯ = 0﷮4﷮5﷮−1﷯﷯ = 0 – 20 = – 20 Minor of a32 = M32 = 1﷮0﷮4﷮3﷮5﷮−1﷮0﷮1﷮2﷯﷯ = 1﷮4﷮3﷮−1﷯﷯ = – 1 – 12 = – 13 Minor of a33 = M33 = 1﷮0﷮4﷮3﷮5﷮−1﷮0﷮1﷮2﷯﷯ = 1﷮0﷮3﷮5﷯﷯ = 5 – 0 = 5 Cofactor of a11 = A11 = ( – 1)1+1 M11 = ( – 1)2 × 11= 1 × 11 = 11 Cofactor of a12 = A12 = ( – 1)1+2 M12 = ( – 1)3 . 6 = ( – 1)6 = – 6 Cofactor of a13 = A13 = ( – 1)1+3 M13 = ( – 1)4 . 3 = (1) 3 = 3 Cofactor of a21 = A21 = ( – 1)2+1 M21 = ( – 1)3 . ( – 4) = ( – 1) ( – 4) = 4 Cofactor of a22 = A22 = ( – 1)2+2 M22 = ( – 1)4 . (2) = (1) . (2) = 2 Cofactor of a23 = A23 = ( – 1)2 + 3 M23 = ( – 1)5 (1) = ( – 1) (1) = – 1 Cofactor of a31= A31 = ( – 1)3 + 1 M31 = (– 1)4 (– 20) = 1 . (– 20) = – 20 Cofactor of a32= A32= (– 1)3 + 2 M32 = (− 1)5 . (– 13) = (– 1) (– 13) = 13 Cofactor of a33 = A33 = ( – 1)3 + 3 M33 = ( – 1)6 . (5) = ( – 1) . (5) = 5

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