# 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:Â  (i) and

Now

Hence

(ii)Â

Then

Thus

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

(i)

(ii)Â

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)Â  It is given that
Therefore

Now

Thus

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

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

Solution:Â  (i) Since
Now

Hence, A is a symmetric matrix.

(ii)Â

Hence, A is a skew symmetric matrix.

Question 8: For the matrix , verify that

(i)Â  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)Â  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)Â  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)Â  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) Â  Â  Â  Â  Â  Â  Â  Â  (B)Â

(C) Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (D)Â

Solution:Thus, the correct option is .

It is given that

Hence,

NOW,

Therefore,

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