NCERT Solutions for Maths Class 12 Exercise 4.4

Hello Students! In this post, you will find the complete NCERT Solutions for Maths Class 12 Exercise 4.4.

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NCERT Solutions for Maths Class 12 Exercise 4.1

NCERT Solutions for Maths Class 12 Exercise 4.2

NCERT Solutions for Maths Class 12 Exercise 4.3

NCERT Solutions for Maths Class 12 Exercise 4.5

NCERT Solutions for Maths Class 12 Exercise 4.4

Find the adjoint of each of the matrices in Questions 1 and 2.

Maths Class 12 Ex 4.4 Question 1.

Solution:
Let A_ij be the cofactor of a_ij in A. Then, the cofactors of elements of A are given by
A₁₁ = (–1)^{1 + 1} (4) = 4; A₁₂ = (–1)^{1 + 2} (3) = –3
A₂₁ = (–1)^{2 + 1} (2)= –2; A₂₂ = (–1)^{2 + 2} (1) = 1
Adj A = Transpose of 

Maths Class 12 Ex 4.4 Question 2.

Solution:

Verify A (adj A) = (adj A) A = |A| I in Questions 3 and 4.

Maths Class 12 Ex 4.4 Question 3.

Solution:

Maths Class 12 Ex 4.4 Question 4.

Solution:
Here, A₁₁ = 0, A₁₂ = –11, A₁₃ = 0, A₂₁ = –3, A₂₂ = 1, A₂₃ = 1, A₃₁ = –2, A₃₂ = 8, A₃₃ = –3

Find the inverse of each of the matrices (if it exists) given in Questions 5 to 11.

Maths Class 12 Ex 4.4 Question 5.

Solution:

So, A is a non-singular matrix and therefore its inverse exists. Let c_ij be the cofactor of a_ij in A. Then, the cofactors of elements of A are given by

Maths Class 12 Ex 4.4 Question 6.

Solution:

So, A is a non-singular matrix and therefore its inverse exists. Let c_ij be the cofactor of a_ij in A. Then, the cofactors of elements of A are given by

C₁₁ = 2, C₁₂ = 3, C₂₁ = –5, C₂₂ = –1 

Maths Class 12 Ex 4.4 Question 7.

Solution:
|A| = 1(10 – 0) = 10
So, A is a non-singular matrix and therefore its inverse exists. Let c_ij be the cofactor of a_ij in A. Then, the cofactors of elements of A are given by

C₁₁ = 10, C₁₂ = 0, C₁₃ = 0, C₂₁ = –10, C₂₂ = 5, C₂₃ = 0, C₃₁ = 2, C₃₂ = –4, C₃₃ = 2 

Maths Class 12 Ex 4.4 Question 8.

Solution:

|A| = 1(–3 – 0) = –3
So, A is a non-singular matrix and therefore its inverse exists. Let c_ij be the cofactor of a_ij in A. Then, the cofactors of elements of A are given by

Maths Class 12 Ex 4.4 Question 9.

Solution:

|A| = 2(–1 – 0) – 1(4 – 0) + 3(8 – 7) = –2 – 4 + 3 = –3 
So, A is a non-singular matrix and therefore its inverse exists. Let c_ij be the cofactor of a_ij in A. Then, the cofactors of elements of A are given by

C₁₁ = –1, C₁₂ = –4, C₁₃ = 1, C₂₁ = 5, C₂₂ = 22, C₂₃ = –11, C₃₁ = 3, C₃₂ = 12, C₃₃ = –6  

Maths Class 12 Ex 4.4 Question 10.

Solution:
|A| = 1(8 – 6) + 1(0 + 9) + 2(0 – 6) = 2 + 9 – 12 = –1 ≠ 0
So, A is a non-singular matrix and therefore its inverse exists. Let c_ij be the cofactor of a_ij in A. Then, the cofactors of elements of A are given by

C₁₁ = 2, C₁₂ = –9, C₁₃ = –6, C₂₁ = 0, C₂₂ = –2, C₂₃ = –1, C₃₁ = –1, C₃₂ = 3, C₃₃ = 2

Maths Class 12 Ex 4.4 Question 11.

Solution:
First find |A| = –cos² α – sin² α = –(cos² α + sin² α) = –1 ≠ 0

So, A is a non-singular matrix and therefore its inverse exists. Let c_ij be the cofactor of a_ij in A. Then, the cofactors of elements of A are given by

C₁₁ = –1, C₁₂ = 0, C₁₃ = 0, C₂₁ = 0, C₂₂ = –cos a, C₂₃ = –sin a, C₃₁ = 0, C₃₂ = –sin a, C₃₃ = cos a

Maths Class 12 Ex 4.4 Question 12.

Let , verify that (AB)^-1 = B^-1A^-1.

Solution:

Here, |A| = 15 – 14 = 1 ≠ 0

Maths Class 12 Ex 4.4 Question 13.

If , show that A² – 5A + 7I = 0, hence find A^-1.

Solution:

Maths Class 12 Ex 4.4 Question 14.

For the matrix A = , find the numbers a and b such that A² + aA + bI = 0. 

Hence, find A^-1.

Solution:

Maths Class 12 Ex 4.4 Question 15.

For the matrix  

Show that A³ – 6A² + 5A + 11 I = 0. Hence, find A^-1.

Solution:

Maths Class 12 Ex 4.4 Question 16.

If  

Verify that A³ – 6A² + 9A – 4 I = 0 and hence, find A^-1.

Solution:

Maths Class 12 Ex 4.4 Question 17.

Let A be a non-singular square matrix of order 3 × 3. Then |Adj A| is equal to:
(A) |A|                (B) |A|²             (C) |A|³               (D) 3|A|

Solution:

Dividing by |A|, |Adj A| = |A|²
Hence, option (B) is correct.

Maths Class 12 Ex 4.4 Question 18.

If A is an invertible matrix of order 2, then det (A^-1) is equal to:
(A) det (A)         (B) 1/det (A)         (C) 1             D) 0

Solution:
|A| ≠ 0
⇒ A^-1 exists

⇒ AA^-1 = I
|AA^-1| = |I| = I
⇒ |A||A^-1| = I

|A^-1| = I/|A|

Hence, option (B) is correct.

Related Links:

NCERT Solutions for Maths Class 12 Exercise 4.1

NCERT Solutions for Maths Class 12 Exercise 4.2

NCERT Solutions for Maths Class 12 Exercise 4.3

NCERT Solutions for Maths Class 12 Exercise 4.5