Problem $1$:

Given $[S_{ij}]=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 2\\ 3& 0& 3\end{array}\right],$

evaluate $($a$)S_{ii},($b$)S_{ij}{S}_{ij},($c$)S_{jk}{S}_{jk},($d$)S_{mn}{S}_{nm}$

Problem $2$:

$($a$)$ If ${}_{\text{i}\text{j}\text{k}}{T}_{ij}=0$ show that $T_{ij}=T_{ji}$

$($b$)$ Show that ${}_{ij}{}_{\text{i}\text{j}\text{k}}=0$

Problem $3$:

Using the relation ${}_{\text{i}\text{j}\text{m}}{}_{klm}={}_{ik}{}_{jl}-{}_{il}{}_{jk},$ show that

vec$(a)(vec(b)$vec$(c))=(vec(a)*vec(c))vec(b)-(vec(a)*vec(b))vec(c)$