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Exercise - 3.3

Question-1 :-  Find the transpose of each of the following matrices: matrices

Solution :-
(i)  matrices
    
(ii)  matrices
    
(iii)  matrices
    

Question-2 :-  matrices (i) (A + B)′ = A′ + B′,
(ii) (A – B)′ = A′ – B′

Solution :-
(i)  matrices
    
(ii)  matrices
    

Question-3 :-  matrices (i) (A + B)′ = A′ + B′,
(ii) (A – B)′ = A′ – B′

Solution :-
(i)  matrices
    
(ii)  matrices
    

Question-4 :-  matrices

Solution :-
matrices
    

Question-5 :-  For the matrices A and B, verify that (AB)′ = B′A′, where matrices

Solution :-
(i)  matrices
    
(ii)  matrices
    

Question-6 :-  matrices

Solution :-
(i)  matrices
    
(ii)  matrices
    

Question-7 :-  matrices

Solution :-
(i)  matrices
    
(ii)  matrices
    

Question-8 :-  matrices (i) (A + A′) is a symmetric matrix
(ii) (A – A′) is a skew symmetric matrix

Solution :-
(i)  matrices
    
(ii)  matrices
    

Question-9 :-  matrices

Solution :-
 matrices
    

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

Solution :-
(i)  matrices
    
(ii)  matrices
    
(iii)  matrices
    
(iv)  matrices
    

Question-11 :-  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

Solution :-
  A and B are symmetric matrices, therefore, we have:
  A' = A, B' = B .........(I)
  Consider 
  (AB - BA)' = (AB)' - (BA)' [(A - B)' = A' - B']
             = B'A' - A'B'   [(AB)' = B'A']
             = AB - BA       [By equation I]
  Thus, (AB − BA) is a skew-symmetric matrix.
  The correct answer is A.
    

Question-12 :-  matrices (A) π/6   (B) π/3   (C) π   (D) 3π/2

Solution :-
matrices
  The correct answer is B.
    
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