# NCERT solution class 12 chapter 3 Matrices exercise 3.3 mathematics part 1

## EXERCISE 3.3

#### Question 1:

Find the transpose of each of the following matrices:

(i)  (ii)  (iii)

(i)

(ii)

(iii)

#### Question 2:

If

and, then verify that

(i)

(ii)

We have:

(i)

(ii)

#### Question 3:

If

and, then verify that

(i)

(ii)

(i) It is known that

Therefore, we have:

(ii)

If

and, then find

We know that

#### Question 5:

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

(i)

(ii)

(i)

(ii)

#### Question 6:

If (i) , then verify that

(ii) , then verify that

(i)

(ii)

#### Question 7:

(i) Show that the matrix is a symmetric matrix

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

(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

(i)

Hence,

is a symmetric matrix.

(ii)

Hence,

is a skew-symmetric matrix.

#### Question 9:

Find

and, when

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)

(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:

#### Question 11:

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

A. Skew symmetric matrix B. Symmetric matrix

C. Zero matrix D. Identity matrix

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.