## NCERT Solutions for Class 12 Maths Chapter 4 – Determinants Ex 4.4

#### Question 1:

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

#### Question 2:

(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

#### Question 3:

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

#### Question 4:

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,

#### Question 5:

If  and Aij is Cofactors of aij, then value of Δ is given by