Chapter 3 Matrices Ex 3.3 Solutions
Question - 1 : - Find the transpose ofeach of the following matrices:
Answer - 1 : -
(i) 
(ii) 
(iii) 
Solution
(i) 
(ii) 
(iii) 
Question - 2 : - If 
and
, then verify that
Answer - 2 : -
(i) 
(ii) 
Solution
We have:
(i)
(ii)
Question - 3 : - If 
and
, then verify that
Answer - 3 : -
(i) 
(ii) 
Solution
(i) It is known that
Therefore, we have:
(ii)
Question - 4 : - If 
and
, then find 
Answer - 4 : -
We know that
Question - 5 : - For the matrices A and B,verify that (AB)′ = 
where
Answer - 5 : -
(i) 
(ii) 
Solution
(i)
(ii)
Question - 6 : - If (i) 
, then verify that 
Answer - 6 : -
(ii) 
, then verify that
Solution
(i)
(ii)
Question - 7 : - (i) Show that thematrix 
is a symmetric matrix
Answer - 7 : -
(ii) Show that thematrix 
is a skew symmetric matrix
Solution
(i) We have:
Hence, A isa symmetric matrix.
(ii) We have:
Hence, A isa skew-symmetric matrix.
Question - 8 : - For the matrix
, verify that
Answer - 8 : -
(i) 
is a symmetric matrix
(ii) 
is a skew symmetric matrix
Solution
(i) 
Hence, is a symmetric matrix.
(ii) 
Hence, is a skew-symmetric matrix.
Question - 9 : - Find 
and
, when 
Answer - 9 : -
The given matrix is
Question - 10 : - Express the followingmatrices as the sum of a symmetric and a skew symmetric matrix:
Answer - 10 : -
(i) 
(ii) 
(iii) 
(iv) 
Solution
(i)
Thus, 
is a symmetric matrix.
Thus, 
is a skew-symmetricmatrix.
Representing A asthe sum of P and Q:
(ii)
Thus, is a symmetric matrix.
Thus, is a skew-symmetricmatrix.
Representing A asthe sum of P and Q:
(iii)
Thus, is a symmetric matrix.
Thus, is a skew-symmetricmatrix..
Representing A asthe sum of P and Q:
(iv)
Thus, is a symmetric matrix.
Thus,is a skew-symmetric matrix.
Representing A asthe sum of P and Q:
