## NCERT Solutions for Class 12 Maths Chapter 3 – Matrices Ex 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.