EDIT: I think I found the answer ... for part a) would it be
0 0 1
0 1 0
1 0 0
Not according to my CAS. Look again at the third power it is back to the original.
But try $\displaystyle
\left( {\begin{array}{ccc}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 0 & 0 \\ \end{array} } \right)$
As such can you use $\displaystyle \left| {AB} \right| = \left| A \right|\left| B \right|$ to prove that $\displaystyle \left| {A^k } \right| = \left| A \right|^k $?
If you can, then that answers your question.