Does $(A^T)^2=(A^2)^T$ for $2\times 2$ matrices?

[karush]

View attachment 8980
this is a c/p from my overleaf DE hw

but on proving this I only used a $2\times 2$ matrix

Also I thot $A^2$ meant A(A) a composite but the calculators just multiplied it.typos maybe

 

[Ackbach]

If you think of a matrix as an operator, then matrix multiplication is functionally equivalent to composition. Recall that $(AB)^T=B^T A^T,$ so that
$$\left(A^T\right)^2=A^TA^T=(AA)^T=\left(A^2\right)^T.$$