= Solution
By the definition of <matrix multiplication> and <trace>, with all indices summed from $1$ to $D$,
$$
\operatorname{tr}(PQR)=\sum_{i,j,k}P_{ij}Q_{jk}R_{ki}.
$$
The entries are scalars, so their multiplication commutes. Reordering them and renaming the summed indices gives
$$
\sum_{i,j,k}Q_{jk}R_{ki}P_{ij}=\sum_{j,k,i}Q_{jk}R_{ki}P_{ij}=\operatorname{tr}(QRP).
$$
Equivalently, in <Einstein notation>, $P_{ij}Q_{jk}R_{ki}=Q_{jk}R_{ki}P_{ij}$. This proves the <cyclic property of the trace>.
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