Question 1

Question 1: Find AT, if (i) A = (ii) A =

Accepted Answer

Step-by-step solution provided.

Question 2

Question 2: If A = [aij]3×3 where aij = 2(i – j). Find A and AT. State whether A and AT both are symmetric or skew-symmetric matrices.

Accepted Answer

Hence, A and AT are both skew-symmetric matrices.

Question 3

Question 3: If A = , prove that (AT)T = A.

Accepted Answer

Step-by-step solution provided.

Question 4

Question 4: If A = , prove that AT = A.

Accepted Answer

Step-by-step solution provided.

Question 5

Question 5: If A = , B = , C = , then show that (i) (A + B)T = AT + BT (ii) (A – C)T = AT – CT

Accepted Answer

Step-by-step solution provided.

Question 6

Question 6: If A = and B = , then find CT, such that 3A – 2B + C = I, where I is the unit matrix of order 2.

Accepted Answer

Step-by-step solution provided.

Question 7

Question 7: If A = , B = , then find (i) AT + 4BT (ii) 5AT – 5BT

Accepted Answer

Step-by-step solution provided.

Question 8

Question 8: If A = , B = and C = , verify that (A + 2B + 3C)T = AT + 2BT + 3CT

Accepted Answer

Step-by-step solution provided.

Question 9

Question 9: If A = and B = , prove that (A + BT)T = AT + B.

Accepted Answer

From (1) and (2), (A + BT)T = AT + B.

Question 10

Question 10: Prove that A + AT is symmetric and A – AT is a skew-symmetric matrix, where (i) A = (ii) A =

Accepted Answer

Step-by-step solution provided.

Question 11

Question 11: Express each of the following matrix as the sum of a symmetric and a skew-symmetric matrix: (i) (ii)

Accepted Answer

Step-by-step solution provided.

Question 12

Question 12: If A = and B = , verify that (i) (AB)T = BTAT (ii) (BA)T = ATBT

Accepted Answer

Step-by-step solution provided.

Chapter 2 Matrices Ex 2.4 Solutions