Exercise 3.3

Find the transpose of each of the following matrices: (i) 5 1/2 /   /        −  (ii) 1 1 / 2 3 / −     (iii) 1 5 6 / 3 5 6 / 2 3 1 / −  /       − 

If / 1 2 3 4 1 5 A 5 7 9 and B 1 2 0 2 1 1 1 3 1 − − −       = =       −    , then verify that (i) (A + B)′ = A′ + B′, (ii) (A - B)′ = A′ - B′

If 3 4 1 2 1A 1 2 and B 1 2 30 1   −  ′ = − =        , then verify that (i) (A + B)′ = A′ + B′ (ii) (A - B)′ = A′ - B′

If 2 3 1 0A and B1 2 1 2 − −   ′ = =       , then find (A + 2B)′

For the matrices A and B, verify that (AB) ′ = B′A′, where (i) [ ] A 4 , B 1 2 1    = − = −     (ii) [ ] A 1 , B 1 5 7    = =    

If (i) cos sinA sin cos α α =  − α α  , then verify that A′ A = I (ii) If sin cosA cos sin α α =  − α α  , then verify that A′ A = I

(i) Show that the matrix 1 1 5

5 1 3 / −   = −     is a symmetric matrix. (ii) Show that the matrix 0 1 1

1 1 0 / −   = −    −  is a skew symmetric matrix.

For the matrix 1 5A 6 7  =    , verify that (i) (A + A′) is a symmetric matrix (ii) (A - A′) is a skew symmetric matrix

Find ( )1 A A2 ′+ and ( )1 A A2 ′− , when a b a c b c    = −   − − 

Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) 3 5 / 1 1 /     −  (ii) 6 2 2 / 2 3 1 / 2 1 3 / −   − −   −  (iii) 3 3 1 / 2 2 1 / 4 5 2 / −   − −   − −  (iv) 1 5 / 1 2 /    −  Choose the correct answer in the Exercises 1 1 and 12.

If A, B are symmetric matrices of same order , then AB - BA is a (A) Skew symmetric matrix (B) Symmetric matrix (C) Zero matrix (D) Identity matrix

If cos sinA ,sin cos α − α =  α α  and A + A′ = I, then the value of α is (A) 6 / π (B) 3 / π (C) π (D) 3 / π