I found the eigenvalues and eigenvectors of the matrix $A$, where $$u'(t)=Au$$ is a ODE system with $$A=\left[\begin{array}{ccc} 0 & 3 & 0 \\ -3 & 0 & 4 \\ 0 & -4 & 0 \end{array}\right]$$ But I have to show "without calculus" (as the exercise asks) that the matrix $e^{At}$ is orthogonal and that $||u(t)||^{2}=u_{1}^{2}+u_{2}^{2}+u_{3}^{2}$ is constant.
I really have no idea how can I do it with no calculus.
HINT: $(e^{At})^\top = e^{A^\top t} = e^{-At}$.
Next, to show $\|u(t)\|^2$ is constant, you should consider its derivative, namely, $2u(t)\cdot u'(t)$. So what is $Au(t)\cdot u(t)$?
Completely avoiding calculus ...
I suppose you could use write out $e^{At}$ explicitly in terms of those eigenvalues and eigenvectors and just check brute-force. Similarly, you have $u(t)=\sum\limits_{j=1}^3 e^{\lambda_j t}v_j$ (where $v_j$ are the eigenvectors and $\lambda_j$ the corresponding eigenvalues), so you could brute-force compute $\|u(t)\|^2$ and see it's constant.