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lec3_1

# lec3_1 - inversions and therefore is even • 52413 has 4 1...

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NOTES ON SECTION 3 . 1 MA265, SECTIONS 41, 52 Key words determinant permutation (even/odd) sign inversion Definition 1. A permutation is a rearrangement of the numbers 1 , . . . , n . (In other words, it is a one-one and onto function σ : { 1 , . . . , n } → { 1 , . . . , n } ). Example 2. These are all permutations: 4321 362541 52413 Notice that each number between the largest and the smallest occurs exactly once. So 452541 is not a permutation because 4 occurs twice, and 63421 is not either because 5 is missing. Definition 3. If j 1 , j 2 , . . . , j n is a permutation, then an inversion occurs when a larger number precedes a smaller number. A permutation is odd if the number of inversions is odd, and even if the number of inversions is even. Example 4. Let’s count the inversions in the previous example. 4321 has 3 + 2 + 1 = 6 inversions, and therefore is even. 362541 has 2 + 4 + 1 + 2 + 1 = 10
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Unformatted text preview: inversions, and therefore is even. • 52413 has 4 + 1 + 2 + 0 = 7 inversions, and therefore is odd. Deﬁnition 5. The sign of a permutation is (-1) number of inversion- in other words, it is +1 if the permutation is even and-1 if the permutation is odd. Deﬁnition 6. Let A be a n × n matrix. Then we deﬁne the determinant of A : det A = X permutations j 1 ,...,j n sign · a 1 ,j 1 a 2 ,j 2 ··· a n,j n This deﬁnition may appear confusing, so we will do some examples to see what this formula means in class. This sum is very large (has n ! terms) if n is large, so we will need other ways to compute the determinant, which we will introduce in the next section. The purpose of this formula is to show that there is such a thing as a determinant at all. 1...
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