# Permutation matrix problem

This is such a kind community, thanks again.

I have a permutation problem and don't know how to solve it.

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?

-
Is is homework? What do you know about permutation matrices? – Davide Giraudo Oct 24 '11 at 19:26
Can you please edit your question and give some details on what you mean by "next permutations" ? – user13838 Oct 24 '11 at 19:27
I don't have a 3rd row. It says: write the permutation matrix which corresponds to the permutations listed in P. – Andrew Oct 24 '11 at 19:32
As given here, $P$ describes one permutation, namely the one that maps 1 to 2, 2 to 3, 3 to 1, 4 to 5 and 5 to 4. In cycle notation (if you're more familiar with that) it would be $P=(1\;2\;3)(4\;5)$. – Henning Makholm Oct 24 '11 at 19:37

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 $\pi$?

-
+1 for your more detailed answer:-) – Jack Oct 25 '11 at 2:28

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?

-
Can you please write down the Ppi? – Andrew Oct 24 '11 at 19:42
@Andrew: See en.wikipedia.org/wiki/Permutation_matrix – Jack Oct 24 '11 at 19:47