Vtyjkyuk

If matrices $Ain M_n(mathbb R)$ orthogonal, then $A^2$ can be orthogonal but it does not have to be orthogonal?

My answer that $A^2$ is always orthogonal, I prove that $|A^2x|=|x|$, but is there better solution to prove that?

Yes, it has to be orthogonal. Suppose that $A$ is orthogonal. What this means is that $A^t.A=operatornameId$. But thenbeginalign(A^2)^t.A^2&=(A.A)^t.A.A\&=A^t.A^t.A.A\&=A^t.operatornameId.A\&=A^t.A\&=operatornameId.endalignTherefore, $A^2$ is orthogonal too.

$A$ is orthogonal $ iff$ $A^T=A^-1$

$$ A^-1=A^T implies A^-2=(A^T)^2=(A^2)^T$$

$$ (A^2)^-1=(A^2)^T$$

$A$ is orthogonal $implies$ $A^2$ is orthogonal.

A square matrix is orthogonal if and only if its columns are orthonormal. For column indices $j,k in 1, ldots, n$ we have

$$sum_i=1^n (A^2)_ij(A^2)_ik = sum_i=1^n left(sum_r=1^n A_irA_rjright)left(sum_s=1^n A_isA_skright) = sum_r=1^n sum_s=1^n left(sum_i=1^n A_irA_isright)A_rjA_sk\ = sum_r=1^n sum_s=1^n delta_rsA_rjA_sk = sum_r=1^n A_rjA_rk = delta_jk$$

so $A^2$ is orthogonal.