# EXERCISE 3.3 (Matrix)

**class 12 maths exercise 3.3 solution**

**Question1:** Find the transpose of each of the following matrices:

**(i) **

**(ii) **

**(iii) **

**Solution: (i) **Let

Then

**(ii) **Let

Then

**(iii) **Let

Then

**Question2:** and Then verify that

**(i)**

**(ii)**

**Solution: ****(i)** and

Now

Hence

**(ii) **

Then

Thus

**Question 5:** For the matrices A and B, verify that

**(i) **

**(ii) **

**Solution: (i)** It is given that and

Therefore,

Now,

Thus,

**(ii)** It is given that and

Hence

Therefore,

Now,

and

Therefore,

Thus,

**Question 6: If (i)** Then verify

**(ii)** Then verify

**Solution: (i) **Since

Therefore,

Now,

Thus

**(ii)** It is given that

Therefore

Now

Thus

**Question 7:(i) ** Show that the matrix is a symmetric matrix.

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

**Solution: (i)** Since

Now

Hence, A is a symmetric matrix.

**(ii) **

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.

**Solution: **It is given that

Hence,

**(i)**

Therefore,

Thus (A+A’) is a symmetric matrix.

**(ii) **

Therefore,

Thus, is a skew symmetric matrix.

**Question 9:** Find and ,When

**Solution:** It is given that

Hence,

Now,

Therefore,

Now,

Thus,

**Question 10:** Express the following as the sum of a symmetric and skew symmetric matrix:

**(i) **

**(ii) **

**(iii) **

**(iv) **

**Solution:(i) **Let

Hence,

Now,

+

Let,

P=

Now,

P’=

Thus,

P= is a symmetric matrix.

Now,

Let,

Q=

Now,

is a skew symmetric matrix.

Representing as the sum of P and Q:

**(ii) **Let

Hence,

Now,

.

Let,

Now,

Thus is a symmetric matrix.

Now,

Let,

Thus, is a skew symmetric matrix.

Representing as the sum of and :

**(iii) **Let

Hence,

Now,

.

Let,

Now,

Thus is a symmetric matrix.

Now,

Let,

Thus, is a skew symmetric matrix.

Representing as the sum of and :

**(iv) **Let

Hence,

Now,

Let,

P=

Now,

P’=

Thus,

P= is a symmetric matrix.

Now,

Let,

Q=

Now,

is a skew symmetric matrix.

Representing as the sum of P and Q:

**Question 11:** If are symmetric matrices of the same order, then is a

(A) Skew symmetric matrix

(B) Symmetric matrix

(C) Zero matrix

(D) Identity matrix

**Solution:The Correct option is **.

If and are symmetric matrices of the same order, then

Solution: If andare symmetric matrices of the same order, then

and —(1)

Now consider,

Therefore,

Thus, is a skew symmetric matrix.

**Question 12:** If then ,if the value of is:

(A) (B)

(C) (D)

**Solution:Thus, the correct option is **.

It is given that

Hence,

NOW,

Therefore,

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