Question

If $A$ is a skew symmetric matrix and $n$ is an even positive integer, then $A^n$ is a

• A. symmetric matrix
• B. skew-symmetric matrix
• C. diagonal matrix
• D. none of these.

Answer 1

Answer: (A)

Given $A' = -A \Rightarrow \left(A'\right)^{n} = \left(-A\right)^{n}$
$ \Rightarrow \left(A^{n}\right)' = \left(\left(-1\right)A\right)^{n} =\left(-1\right)^{n}A^{n} $
$ \Rightarrow \left(A^{n}\right)' = A^{n}$ ($\because $ n is even)
$ \Rightarrow A^{n} $ is a symmetric matrix.