The permutation matrix P=(001100010)P = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}P=010001100 corresponds to the cyclic permutation (1→2→3→1)(1 \to 2 \to 3 \to 1)(1→2→3→1). What is det(P)\det(P)det(P)?
det(P)=−1\det(P) = -1det(P)=−1 (3-cycle is odd)
det(P)=1\det(P) = 1det(P)=1 (3-cycle is even)
det(P)=0\det(P) = 0det(P)=0
Cannot be determined from cycle structure alone