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