Permutation matrix problem

Question

Write the permutation matrix $P_{\pi}$ for the next permutations: $$ P= \pmatrix{1&2&3&4&5\\2 &3 &1 &5 &4} $$ What is $\det (P_{\pi})$ without really calculating it?

Answer 1

The "permutation matrix" associated to $\pi$ is the matrix that is obtained from the identity matrix by "swapping columns" according to the permutation $\pi$.

For example, if $$\pi = \left(\begin{array}{cccc} 1 & 2 & 3 & 4\\ 2 & 3 & 1 & 4 \end{array}\right),$$ then the permutation matrix would be the matrix obtained from the identity by moving the first column to the 2nd column position; the second column to the third column position; the third column to the first column position; and leaving the fourth column in the fourth column position. That is, $$P_{\pi}=\left(\begin{array}{cccc} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array}\right).$$

Because $P_{\pi}$ is obtained from the identity by swapping columns, its determinant will be either $1$ or $-1$; it is $1$ if you performed an even number of column exchanges/swaps, and $-1$ if you performed an odd number of column/swaps exchanges.

How does the parity of the number of column exchanges/swaps relate to $\pi$?

Answer 2

Hint: Do you know the parity of the permutation?

What's more, for finding the determinant of a matrix, instead of calculating it by definition, people usually use the properties of the determinant. Have you written down your $P_{\pi}$? Do you see how can one get $P_{\pi}$ by the identity matrix?