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Write Minors and Cofactors of the elements of following determinants:
(i) 
(ii) 
(i) The given determinant is.
Minor of element aij is Mij.
∴M11 = minor of element a11 = 3
M12 = minor of element a12 = 0
M21 = minor of element a21 = −4
M22 = minor of element a22 = 2
Cofactor of aij is Aij = (−1)i + j Mij.
∴A11 = (−1)1+1 M11 = (−1)2 (3) = 3
A12 = (−1)1+2 M12 = (−1)3 (0) = 0
A21 = (−1)2+1 M21 = (−1)3 (−4) = 4
A22 = (−1)2+2 M22 = (−1)4 (2) = 2
(ii) The given determinant is.
Minor of element aij is Mij.
∴M11 = minor of element a11 = d
M12 = minor of element a12 = b
M21 = minor of element a21 = c
M22 = minor of element a22 = a
Cofactor of aij is Aij = (−1)i + j Mij.
∴A11 = (−1)1+1 M11 = (−1)2 (d) = d
A12 = (−1)1+2 M12 = (−1)3 (b) = −b
A21 = (−1)2+1 M21 = (−1)3 (c) = −c
A22 = (−1)2+2 M22 = (−1)4 (a) = a
(i) 
(ii) 
(i) The given determinant is.
By the definition of minors and cofactors, we have:
M11 = minor of a11= 
M12 = minor of a12= 
M13 = minor of a13 = 
M21 = minor of a21 =
M22 = minor of a22 =
M23 = minor of a23 = 
M31 = minor of a31= 
M32 = minor of a32 =
M33 = minor of a33 =
A11 = cofactor of a11= (−1)1+1 M11 = 1
A12 = cofactor of a12 = (−1)1+2 M12 = 0
A13 = cofactor of a13 = (−1)1+3 M13 = 0
A21 = cofactor of a21 = (−1)2+1 M21 = 0
A22 = cofactor of a22 = (−1)2+2 M22 = 1
A23 = cofactor of a23 = (−1)2+3 M23 = 0
A31 = cofactor of a31 = (−1)3+1 M31 = 0
A32 = cofactor of a32 = (−1)3+2 M32 = 0
A33 = cofactor of a33 = (−1)3+3 M33 = 1
(ii) The given determinant is.
By definition of minors and cofactors, we have:
M11 = minor of a11= 
M12 = minor of a12= 
M13 = minor of a13 = 
M21 = minor of a21 = 
M22 = minor of a22 = 
M23 = minor of a23 = 
M31 = minor of a31= 
M32 = minor of a32 = 
M33 = minor of a33 = 
A11 = cofactor of a11= (−1)1+1 M11 = 11
A12 = cofactor of a12 = (−1)1+2 M12 = −6
A13 = cofactor of a13 = (−1)1+3 M13 = 3
A21 = cofactor of a21 = (−1)2+1 M21 = 4
A22 = cofactor of a22 = (−1)2+2 M22 = 2
A23 = cofactor of a23 = (−1)2+3 M23 = −1
A31 = cofactor of a31 = (−1)3+1 M31 = −20
A32 = cofactor of a32 = (−1)3+2 M32 = 13
A33 = cofactor of a33 = (−1)3+3 M33 = 5
Using Cofactors of elements of second row, evaluate
.
The given determinant is
.
We have:
M21 = 
∴A21 = cofactor of a21 = (−1)2+1 M21 = 7
M22 = 
∴A22 = cofactor of a22 = (−1)2+2 M22 = 7
M23 = 
∴A23 = cofactor of a23 = (−1)2+3 M23 = −7
We know that Δ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.
∴Δ = a21A21 + a22A22 + a23A23 = 2(7) + 0(7) + 1(−7) = 14 − 7 = 7
Using Cofactors of elements of third column, evaluate
The given determinant is
.
We have:
M13 = 
M23 = 
M33 = 
∴A13 = cofactor of a13 = (−1)1+3 M13 = (z − y)
A23 = cofactor of a23 = (−1)2+3 M23 = − (z − x) = (x − z)
A33 = cofactor of a33 = (−1)3+3 M33 = (y − x)
We know that Δ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.
Hence, 
If 
and Aij is Cofactors of aij, then value of Δ is given by
We know that:
Δ = Sum of the product of the elements of a column (or a row) with their corresponding cofactors
∴Δ = a11A11 + a21A21 + a31A31
Hence, the value of Δ is given by the expression given in alternative D.
The correct answer is D.
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