Which of the following statements is/are true If A and B are two skew-symmetric matrices of order n then 1. A ⋅ B is a skew symmetric matrix when AB = - BA 2. A ⋅ B is a symmetric matrix when AB = BA
Which of the following statements is/are true If A and B are two skew-symmetric matrices of order n then 1. A ⋅ B is a skew symmetric matrix when AB = - BA 2. A ⋅ B is a symmetric matrix when AB = BA Correct Answer Both 1 and 2
Concept:
- Symmetric Matrix: Any real square matrix A = (aij) is said to be symmetric matrix if and only if aij = aji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = A’ then A is said to be a symmetric matrix.
- Skew-symmetric Matrix: Any real square matrix A = (aij) is said to be skew-symmetric matrix if and only if aij = - aji, ∀ i and j or in other words we can say that if A is a real square matrix such that A =- A’ then A is said to be a skew-symmetric matrix.
- (A ± B)' = A' ± B'
- (A ⋅ B)' = B' ⋅ A'
Calculation:
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)' = 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)' = B ⋅ A
⇒ (A ⋅ B)' = (A ⋅ B)--------------(∵ AB = BA)
Hence, statement 2 is also true.
