Question

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

Answer 1

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

\[ \left( A^n \right)^T = \left( A^T \right)^n \left[ \text{For all  n }\in N \right]\] 

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

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

`( A^n \right)^T = A^n , if n  is  even or - A^n , if n   is  odd.`

Hence ,  `A^n` symmetric when n is an even natural number.