Question

If A is a skew-symmetric matrix and n is an odd natural number, write whether Aⁿ is symmetric or skew-symmetric or neither of the two.

Answer 1

`If A  is  a  skew - symmetric  matrix, then  A^T = - A`

`( A^n )^T = ( A^T)^n [  " For "all  n  ∈  N ]`

\[ \Rightarrow \left( A^n \right)^T = \left( - A \right)^n \left[ \because A^T = - A \right]\] 

\[ \Rightarrow \left( A^n \right)^T = \left( - 1 \right)^n A^n \] 

\[ \Rightarrow \left( A^n \right)^T = A^n , \text{if n is even or - A^n , if n is odd} .\]

Hence, `( A)^n `is skew-symmetric when n is an odd natural number.