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Chapter 8: Problem 16

Determine the parity (sign) of the permutation \(\sigma=364152\)

Short Answer

Expert verified
The parity of the permutation \(\sigma = 364152\) is even, as it has 10 inversions, which is an even number.

Step by step solution

01

Write the permutation in two-line notation

To work with the permutation more easily, let's represent it in two-line notation. In this case, the permutation is: \[ \sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 6 & 4 & 1 & 5 & 2 \end{pmatrix} \]
02

Count the inversions

Now let's count the number of inversions in the permutation. For each pair of positions \(i < j\), check if \(\sigma(i) > \sigma(j)\). If yes, that's an inversion. For i = 1: - 1: Compared to 6 (j = 2), 4 (j = 3), 1 (j = 4), 5 (j = 5), and 2 (j = 6), we have 3 inversions (6, 4, 2). For i = 2: - 6: Compared to 4 (j = 3), 1 (j = 4), 5 (j = 5), and 2 (j = 6), we have 3 inversions (4, 1, 2). For i = 3: - 4: Compared to 1 (j = 4), 5 (j = 5), and 2 (j = 6), we have 2 inversions (1, 2). For i = 4: - 1: Compared to 5 (j = 5) and 2 (j = 6), we have 1 inversion (2). For i = 5: - 5: Compared to 2 (j = 6), we have 1 inversion (2). This means, in total, we have 3 + 3 + 2 + 1 + 1 = 10 inversions.
03

Determine the parity

Now that we have counted the number of inversions (10), we can determine the parity of the permutation. Since 10 is an even number, the permutation has an even number of inversions, so the parity of permutation \(\sigma = 364152\) is even.

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Most popular questions from this chapter

Let \(V\) be the space of \(m\) -square matrices (as above), and suppose \(D: V \rightarrow K\). Show that the following weaker statement is equivalent to \(D\) being alternating: $$D\left(A_{1}, A_{2}, \ldots, A_{n}\right)=0 \quad \text { whenever } \quad A_{i}=A_{i+1} \text { for some } i$$

Let \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 0 & 7 & 8\end{array}\right],\) and let \(S_{k}\) denote the sum of its principal minors of order \(k\). Find \(S_{k}\) for (a) \(k=1\) (b) \(k=2\) (c) \(k=3\)

Let \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \cdot\) (a) \(\quad\) Find adj \(A\) (b) Show that \(\operatorname{adj}(\operatorname{adj} A)=A\) (c) When does \(A=\operatorname{adj} A ?\)

For \(k=1,2,3,\) find the sum \(S_{k}\) of all principal minors of order \(k\) for (a) \(A=\left[\begin{array}{rrr}1 & 3 & 2 \\ 2 & -4 & 3 \\ 5 & -2 & 1\end{array}\right]\) (b) \(B=\left[\begin{array}{rrr}1 & 5 & -4 \\ 2 & 6 & 1 \\ 3 & -2 & 0\end{array}\right]\) (c) \(C=\left[\begin{array}{rrr}1 & -4 & 3 \\ 2 & 1 & 5 \\ 4 & -7 & 11\end{array}\right]\)

Prove Theorem 8.5: Let \(A\) be an \(n\) -square matrix. Then the following are equivalent: (i) \(A\) is invertible, (ii) \(A X=0\) has only the zero solution, (iii) \(\operatorname{det}(A) \neq 0\)
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