Find the effect of a rotation by $\pi / 2$ about the $z$-axis on the tensor

$$\left(\begin{array}{lll} S_{11} & S_{12} & S_{13} \\ S_{21} & S_{22} & S_{23} \\ S_{31} & S_{32} & S_{33} \end{array}\right)$$

Hence show that the most general isotropic tensor of rank 2 is $\lambda \delta_{i j}$, where $\lambda$ is an arbitrary scalar.

Prove that there is no non-zero isotropic vector, and write down without proof the most general isotropic tensor of rank 3 .

Deduce that if $T_{i j k l}$ is an isotropic tensor then the following results hold, for some scalars $\mu$ and $\nu$ : (i) $\epsilon_{i j k} T_{i j k l}=0$; (ii) $\delta_{i j} T_{i j k l}=\mu \delta_{k l}$; (iii) $\epsilon_{i j m} T_{i j k l}=\nu \epsilon_{k l m}$.

Verify these three results in the case $T_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}$, expressing $\mu$ and $\nu$ in terms of $\alpha, \beta$ and $\gamma$.