Class 12 RD Sharma Solutions- Chapter 5 Algebra of Matrices – Exercise 5.4

Question 1: Let A = and B = verify that

(i) (2A)^T = 2A^T

(ii) (A + B)^T = A^T + B^T

(iii) (A − B)^T = A^T − B^T

(iv) (AB)^T = B^T A^T

Solution:

(i) Given: A = and B = 

Assume,

(2A)^T = 2A^T

Substitute the value of A

L.H.S = R.H.S

Hence, proved.

(ii) Given: A = and B = 

Assume,

(A+B)^T = A^T + B^T

L.H.S = R.H.S

Hence, proved.

(iii) Given: A= and B= 

Assume,

(A − B)^T = A^T − B^T

L.H.S = R.H.S

Hence, proved

(iv) Given: A = and B = 

Assume,

(AB)^T = B^TA^T

Therefore, (AB)^T = B^TA^T

Hence, proved.

Question 2: A = and B =  Verify that (AB)^T = B^TA^T

Solution:

Given: A = and B = 

Assume,

(AB)^T = B^TA^T

L.H.S = R.H.S

Hence proved

Question 3: Let A = and B = 

Find A^T, B^T and verify that

(i) (A + B)^T = A^T + B^T

(ii) (AB)^T = B^TA^T

(iii) (2A)^T = 2A^T

Solution:

(i) Given: A = 

and B = 

Assume

(A + B)^T = A^T + B^T

L.H.S = R.H.S

Hence proved

(ii) Given: A = and B = 

Assume,

(AB)^T = B^TA^T

L.H.S =R.H.S

Hence proved

(iii) Given: A = and B = 

Assume,

(2A)^T = 2A^T

L.H.S = R.H.S

Hence proved

Question 4: if A = , B = , verify that (AB)^T = B^TA^T

Solution:

Given: A = and B = 

Assume,

(AB)^T = B^TA^T

L.H.S = R.H.S

Hence proved

Question 5: If A = and B = , find (AB)T

Solution:

Given: A = and B = 

Here we have to find (AB)^T

Hence,

(AB)^T = 

Question 6: 

(i) For two matrices A and B,   verify that (AB)^T = B^TA^T

Solution:

Given,

(AB)^T = B^TA^T

⇒ 

⇒ 

⇒ 

⇒ 

⇒ L.H.S = R.H.S

Hence,

(AB)^T = B^TA^T

(ii) For the matrices A and B, verify that (AB)^T = B^TA^T, where

Solution:

Given,

(AB)^T = B^TA^T

⇒ 

⇒ 

⇒ 

⇒ 

⇒ L.H.S = R.H.s

So,

(AB)^T = B^TA^T

Question 7: Find , A^T – B^T

Solution:

Given that 

We need to find A^T – B^T.

Given that, 

 

Let us find A^T – B^T

⇒ 

⇒ 

⇒ 

Question 8: If , then verify that A’A = 1

Solution:

⇒ 

⇒ 

⇒ 

Hence,we have verified that A’A = I

Question 9: , then verify that A’A = I

Solution:

Hence, we have verified that A’A = I

Question 10: If l_i, m_i, n_i ; i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AA^T = I,

Where 

Solution:

Given,

l_i, m_i, n_i are direction cosines of three mutually perpendicular vectors

⇒ 

And,

Given,

= I

Hence,

AA^T = I