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**Unformatted text preview: **ther using transpositions, but they
will either all be an odd number or all an even number. For example, we could do
(12) 123 → 213 → 231 → 132 → 312 → 321 to get from 123 to 321, or we can simply do
(13) 123 → 321. One way took ﬁve transpositions the other took one, and both one and ﬁve are odd numbers. No matter what
sequence of transpositions you use to get from 123 to 321, you’ll always have an odd number of transpositions.
Even and odd diagonals. If you look carefully at all the positive terms in the determinant, they all run along
a diagonal oriented from the upper-left to lower-right. The negative terms go with diagonals that run from
lower-left to upper-right. This pattern doesn’t extend easily to higher dimensions.
Homework 19
1. Find the following determinants. a. 2 1
5 2 b. 4 1
3 2 c. 5 2
5 2 d. 1 2 1
3 2 1
1 1 1 MA 3280 Lecture 19 - Determinants e. 3 1 2 1
1 2 1
1 1 1 2.
Find a sequence of transpositions that gets you from the standard ordering to the given permutation.
Is the permutation even or odd?
a. 132 b. 321 c. 1243 d. 2431 e. 4321 MA 3280 Lecture 19 - Determinants Answers:
1a) −1. b) 5. c) 0. d) −2. e) 0.
2 There are inﬁnitely many sequences of transpositions that will work. I’m just giving one.
a) 123 → 132, odd.
b) 123 → 132 → 312 → 321, odd.
c) 1234 → 1243, odd.
d) 1234 → 2134 → 2431, even.
e) 1234 → 4231 → 4321, even. 4...

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