Show that A² is symmetric , if either A is symmetric or A is skew symmetric

Question

Sir it is from Bsc 1 semester book of maths

Answer 1

Case 1) If A is symmetric => A = AT, where AT is transpose of A

So, (AT)² = (AT*AT) = (AA) = A²

Case 2) If A is skew-symmetric => A = -AT

So, (-AT)² = (-AT*-AT)=AA = A²

So, from both case 1 and 2 we can say that A² is symmetric if A is either symmetric or skew-symmetric.