UPSC NDA Math Model Paper 3

Given: A and B are two skew-symmetric matrices of order n
Statement 1: A ⋅ B is a skew symmetric matrix when AB = - BA
Let's find out transpose of (A ⋅ B)
⇒ (A ⋅ B)' = B' ⋅ A'
∵ A and B are two skew - symmetric matrices of order n i.e A'
= A and B' = B
⇒ (A ⋅ B)' = B ⋅ A
⇒ (A ⋅ B)' = - (A ⋅ B)--------------(∵ AB = - BA)
Hence, statement 1 is true.
Statement 2: A ⋅ B is a symmetric matrix when AB = BA
Let's find out transpose of (A ⋅ B)
⇒ (A ⋅ B)' = B' ⋅ A'
∵ A and B are two skew - symmetric matrices of order n i.e A' = A and B' = B
⇒ (A ⋅ B)' = B ⋅ A
⇒ (A ⋅ B)' = (A ⋅ B)--------------(∵ AB = BA)
Hence, statement 2 is also true.