Question

Show that positive odd integral powers of a skew-symmetric matrix are skew-symmetric
and positive even integral powers of a skew-symmetric matrix are symmetric.
(Please do not solve in RD Sharma method, I didn't understand that one)​

Answer 1

Step-by-step explanation:

Here, Ais a skew-symmetric matrix.

∴A^T = −A.

Now, We know, (A^n)^T=(A^T)^n

⇒(A^n)^T=(−A)^n →(1)

When nn is an odd natural number,

⇒(A^n)^T=−A^n

It means, A^n will be a skew-symmetric matrix for a odd number n, if A is a skew-symmetric matrix.

When nn is an even natural number,

⇒(A^n)^T=(−A)^n

⇒(A^n)^T=A^n

It means, A^nwill be a symmetric matrix for an even number n, if Ais a skew-symmetric matrix.