Which of the following statements is/are true: If A is a skew-symmetric matrix of order n then 1. A2m is a symmetric matrix where m is a positive integer. 2. A2m + 1 is a skew - symmetric matrix where m is a positive integer.

Which of the following statements is/are true: If A is a skew-symmetric matrix of order n then 1. A2m is a symmetric matrix where m is a positive integer. 2. A2m + 1 is a skew - symmetric matrix where m is a positive integer. 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'
• (An)' = (A')n
• (k × A)’ = k × A’

Calculation:

Given: A is a skew-symmetric matrix of order n

Statement 1: A2m is a symmetric matrix where m is a positive integer.

Let's find out the transpose of the matrix A^2m 

⇒ (A^2m)' = (A')^2m

∵ A is a skew-symmetric matrix of order n i.e A' = - A

⇒ (A2m)' = A^2m

So, A^2m is a symmetric matrix.

Hence statement 1 is true.

Statement 2: A^{2m + 1} is a skew - symmetric matrix where m is a positive integer.

Let's find out the transpose of the matrix A^{2m + 1} 

⇒ (A^{2m + 1})' = (A')^{2m + 1}

∵ A is a skew-symmetric matrix of order n i.e A' = - A

⇒ (A2m + 1)' = (A')^2m ⋅ A' = - A^{2m + 1}

So, A^{2m + 1} is a skew symmetric matrix.

Hence statement 2 is also true.