Question

Prove that all +ve integral powers of a symmetric matrix are symmetric.

Answer 1

Let $A$ be symmetric i.e., $A^{\prime }=A$
$A^P=AAA.....p$ times
or $(A^P{)}^{\prime }=(AAA....ptimes{)}^{\prime }$
$=(A^{\prime }{A}^{\prime }{A}^{\prime }.....ptimes)$
$=(AAA......ptimes)=A^P$ $(\because$ $A^{\prime }=A)$
Hence $A^P$ is symmetric