Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
Is is homework? What do you know about permutation matrices? –  Davide Giraudo Oct 24 '11 at 19:26
2  
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

2 Answers 2

up vote 5 down vote accepted

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 $\pi$?

share|improve this answer
    
+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?

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.