NCERT Solutions Class 12 maths Chapter-4 (Determinants)Exercise 4.4

NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-4 (Determinants)Exercise 4.4 This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter.
We have given the answers of all the questions of  NCERT Board Mathematics Textbook in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination.

Exercise 4.4

Write minors and cofactors of the elements of the following determinants: 

Question 1. 

(i) 

(ii) 

Solution: 

(i) 

Finding minors of the elements of the determinant: 

Let us assume M_ij is Minors of elements a_ij

M₁₁ = Minor of elements a₁₁ = 3

M₁₂ = Minor of elements a₁₂ = 0

M₂₁ = Minor of elements a₂₁ = −4

M₂₂ = Minor of elements a₂₂ = 2

Finding cofactor of a_ij

Let us assume cofactor of a_ij is A_ij M_ij

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) 

Finding minors of the elements of the determinant:

Let us assume M_ij is Minors of elements a_ij

M₁₁ = Minor of element a₁₁ = d

M₁₂ = Minor of elements a₁₂ = b

M₂₁ = Minor of elements a₂₁ = c

M₂₂ = Minor of elements a₂₂ = a

Finding cofactor of a_ij

Let us assume cofactor of a_ij is A_ij, which is (−1)^i+j M_ij

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)

Solution: 

(i) 

Let us find the Minors and cofactors of the elements:

Assume, Mij is minor of element aij and Aij is cofactor of aij

M11 = Minor of elements a11 == 1 − 0 = 1 and A11 = 1

M12 = Minor of elements a12 == 0 − 0 = 0 and A12 = 0

M13 = Minor of elements a13 == 0 − 0 = 0 and A13 = 0

M21 = Minor of elements a21 == 0 − 0 = 0 and A21 = 0

M22 = Minor of elements a22 == 1 − 0 = 1 and A22 = 1

M23 = Minor of elements a23 == 0 − 0 = 0 and A23 = 0

M31 = Minor of elements a31 == 0 − 0 = 0 and A31 = 0

M32 = Minor of elements a32 == 0 − 0 = 0 and A32 = 0

M33 = Minor of elements a33 == 1 − 0 = 1 and A33 = 1

(ii) 

Let us find the Minors and cofactors of the elements:

Assume, Mij is minor of element aij and Aij is cofactor of aij

M11 = Minor of elements a11 == 10 − (−1) = 11 and A11 = 11

M12 = Minor of elements a12 == 6 − 0 = 6 and A12 = −6

M13 = Minor of elements a13 == 3 − 0 = 3 and A13 = 3

M21 = Minor of elements a21 == 0 − 4 = −4 and A21 = 4

M22 = Minor of elements a22 == 2 − 0 = 2 and A22 = 2

M23 = Minor of elements a23 == 1 − 0 = 1 and A23 = −1

M31 = Minor of elements a31 == 0 − 20 = −20 and A31 = −20

M32 = Minor of elements a32 == −1 − 12 = −13 and A32 = 13

M33 = Minor of elements a33 == 5 − 0 = 5 and A33 = 5

Question 3. Using Cofactors of elements of second row, evaluate △?

Solution: 

Finding the Cofactors of elements of second row:

A21 = Cofactor of elements a21 = (−1)2+1  = (−1)3 (9 − 16) = 7

A22 = Cofactor of elements a22 = (−1)2+2  = (−1)4 (15 − 8) = 7

A23 = Cofactor of elements a23 = (−1)2+3  = (−1)5 (10 − 3) = 7

Now, △ = a21 A21 + a22 A22 + a23 A23 = 14 + 0 − 7 = 7

Question 4. Using Cofactors of elements of third column, evaluate △?

Solution: 

Finding the Cofactors of elements of third column:

A13 = Cofactor of elements a13 = (−1)1+3  = (−1)4 (z − y) = −y

A23 = Cofactor of elements a23 = (−1)2+3  = (−1)5 (z − x) = x − z

A33 = Cofactor of elements a33 = (−1)3+3  = (−1)6 (y − x) = y − x

Now, △ = a13 A13 + a23 A23 + a33 A33

= yz (z − y) + zx (x − z) + xy (y − x)

= (yz2 − y2z) + (xy2 − xz2) + (xz2 − x2y)

= (y − z)[−yz + x(y + z) − x2]

= (y − z)[−yz + x (z − x) + x (z − x)]

= (x − y)(y − x)(z − x)

Question 5. If  and A_ij is cofactor of a_ij then value of △ is given by:

(A) a₁₁A₃₁ + a₁₂A₃₂ + a₁₃A₃₃

(B) a₁₁A₁₁ + a₁₂A₂₁ + a₁₃A₃₁

(C) a₂₁A₁₁ + a₂₂A₁₂ + a₂₃A₁₃

(D) a₁₁A₁₁ + a₂₁A₂₁ + a₃₁A₃₁

Solution: 

Option (D) is correct.

Chapter-4 (Determinants)

• Exercise 4.1
• Exercise 4.2
• Exercise 4.3
• Exercise 4.5
• Exercise 4.6