NCERT Solutions for Class 12 Maths Chapter 3 – Matrices Ex 3.3

NCERT Solutions for Class 12 Maths Chapter 3 – Matrices Ex 3.3

Page No 88:

Question 1:

Find the transpose of each of the following matrices:

(i)  (ii)  (iii) 

Answer:

(i) 

(ii) 

(iii) 

Question 2:

If

and, then verify that

(i) 

(ii) 

Answer:

We have:

(i)

(ii)

Question 3:

If

and, then verify that

(i) 

(ii) 

Answer:

(i) It is known that

Therefore, we have:

(ii)

Question 4:

If

and, then find 

Answer:

We know that

Question 5:

For the matrices A and B, verify that (AB)′ =  where

(i) 

(ii) 

Answer:

(i)

(ii)

Page No 89:

Question 6:

If (i) , then verify that 

(ii) , then verify that 

Answer:

(i)

(ii)

Question 7:

(i) Show that the matrix is a symmetric matrix

(ii) Show that the matrix is a skew symmetric matrix

Answer:

(i) We have:

Hence, A is a symmetric matrix.

(ii) We have:

Hence, A is a skew-symmetric matrix.

Question 8:

For the matrix, verify that

(i)  is a symmetric matrix

(ii)  is a skew symmetric matrix

Answer:

(i) 

Hence,

 is a symmetric matrix.

(ii) 

Hence,

is a skew-symmetric matrix.

Question 9:

Find

and, when 

Answer:

The given matrix is

Question 10:

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

(i) 

(ii) 

(iii) 

(iv) 

Answer:

(i)

Thus,  is a symmetric matrix.

Thus,  is a skew-symmetric matrix.

Representing A as the sum of P and Q:

(ii)

Thus,  is a symmetric matrix.

Thus,  is a skew-symmetric matrix.

Representing A as the sum of P and Q:

(iii)

Thus,  is a symmetric matrix.

Thus,  is a skew-symmetric matrix.

Representing A as the sum of P and Q:

(iv)

Thus,  is a symmetric matrix.

Thus, is a skew-symmetric matrix.

Representing A as the sum of P and Q:

Page No 90:

Question 11:

If A, B are symmetric matrices of same order, then AB − BA is a

A. Skew symmetric matrix B. Symmetric matrix

C. Zero matrix D. Identity matrix

Answer:

The correct answer is A.

A and B are symmetric matrices, therefore, we have:

Thus, (AB − BA) is a skew-symmetric matrix.

Question 12:

If, then, if the value of α is

A.  B. 

C. π D. 

Answer:

The correct answer is B.

Comparing the corresponding elements of the two matrices, we have: