Exercise 4.4
Write Minors and Cofactors of the elements of following determinants:
Solun:- Cofactor = (-1)i+jMij
⇒ Minor of 2 = |3| = 3 and Cofactor of 2 = (-1)1+1|3| = 3
⇒ Minor of -4 = |0| = 0 and Cofactor of -4 = (-1)1+2|0| = 0
⇒ Minor of 0 = |-4| = -4 and Cofactor of 0 = (-1)2+1|-4| = 4
⇒ Minor of 3 = |2| = 2 and Cofactor of 3 = (-1)2+2|2| = 2
Solun:- Cofactor = (-1)i+jMij
⇒ Minor of a = |d| = d and Cofactor of a = (-1)1+1|d| = d
⇒ Minor of c = |b| = b and Cofactor of c = (-1)1+2|b| = -b
⇒ Minor of b = |c| = c and Cofactor of b = (-1)2+1|c| = -c
⇒ Minor of d = |a| = a and Cofactor of d = (-1)2+2|a| = a
⇒ Cofactor = (-1)i+jMij
⇒ Cofactor of 1 = (-1)1+1 M11 = 1
⇒ Cofactor of 0 = (-1)1+2 M12 = 0
⇒ Cofactor of 0 = (-1)1+3 M13 = 0
⇒ Cofactor of 0 = (-1)2+1 M21 = 0
⇒ Cofactor of 1 = (-1)2+2 M22 = 1
⇒ Cofactor of 0 = (-1)2+3 M23 = 0
⇒ Cofactor of 0 = (-1)3+1 M31 = 0
⇒ Cofactor of 0 = (-1)3+2 M32 = 0
⇒ Cofactor of 1 = (-1)3+3 M33 = 1
⇒ Cofactor = (-1)i+jMij
⇒ Cofactor of 1 = (-1)1+1 M11 = 11
⇒ Cofactor of 0 = (-1)1+2 M12 = -6
⇒ Cofactor of 4 = (-1)1+3 M13 = 3
⇒ Cofactor of 3 = (-1)2+1 M21 = 4
⇒ Cofactor of 5 = (-1)2+2 M22 = 2
⇒ Cofactor of -1 = (-1)2+3 M23 = -1
⇒ Cofactor of 0 = (-1)3+1 M31 = -20
⇒ Cofactor of 1 = (-1)3+2 M32 = 13
⇒ Cofactor of 2 = (-1)3+3 M33 = 5
3. Using Cofactors of elements of second row, evaluate:
Solun:- We know the sum of the product of elements of any row (or column) with their corresponding cofactors is a determinant value.
Cofactor of second row:-
⇒ Δ = a21A21+a22A22+a23A23 = 14+0-7 = 7
4. Using Cofactors of elements of third column, evaluate:
Solun:- We know the sum of the product of elements of any row (or column) with their corresponding cofactors is a determinant value.
Cofactor of third column:-
⇒ Δ = a13A13+a23A23+a33A33 = yz(z-y)+zx(x-z)+xy(y-x)
⇒ Δ = yz2-y2z+zx2-xz2+xy2-x2y
⇒ Δ = (yz2-y2z)+(-xz2+xy2)+(zx2-x2y)
⇒ Δ = yz(z-y)+x(y2-z2)-x2(y-z)
⇒ Δ = -yz(y-z)+x(y-z)(y+z)-x2(y-z)
⇒ Δ = (y-z){-yz+x(y+z)-x2}
⇒ Δ = (y-z){-yz+xy+xz-x2}
⇒ Δ = (y-z){y(x-z)-x(x-z)}
⇒ Δ = (y-z)(x-z)(y-x)
⇒ Δ = (x-y)(y-z)(z-x)
(A) a11A31+a12A32+a13A33
(B) a11A11+a12A21+a13A31
(C) a21A11+a22A12+a23A13
(D) a11A11+a21A21+a31A31
Solun:- We know the sum of the product of elements of any row (or column) with their corresponding cofactors is a determinant value.
Then Answer is………….D.
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