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= MA 265 Chapter 3 Sections 3.1-3.2 = | = MA 265 Chapter 3 Sections 3.1-3.2 = | ||
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= By: Daniel Ford = | = By: Daniel Ford = | ||
| − | = What are determinants? = | + | = <u>What are determinants?</u> = |
To understand determinants, you must first know about permutations. | To understand determinants, you must first know about permutations. | ||
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| + | == <u>'''Permutations'''</u> == | ||
If D = {1, 2,....,n} a set of integers from 1 to n in ascending order, then a permutation is the rearrangement of an integer in D. | If D = {1, 2,....,n} a set of integers from 1 to n in ascending order, then a permutation is the rearrangement of an integer in D. | ||
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If D = {6, 7, 8, 9}, then 7689 would be a permutation of D. This corresponds to the function f: D→ D defined by | If D = {6, 7, 8, 9}, then 7689 would be a permutation of D. This corresponds to the function f: D→ D defined by | ||
| − | | + | f(1) = 6<br> f(2) = 7<br> f(3) = 8<br> f(4) = 9<br>Then after permutation,<br> f(1) = 7<br> f(2) = 6<br> f(3) = 8<br> f(4) = 9 |
| − | + | The reason that this is important to know is that the total number of permutations can be even or odd. So in the permutation 4321, where 4 precedes 3, 4 precedes 1, 4 prececedes 2, 3 precedes 1, and 3 precedes 2. Thus the total number of inversions in this premutation is 5, which would make 4321, odd. This determines what sign you put in front, so if it is odd, then you put a negative sign (-) in front; and if it is even, then you put a postive sign (+) in front. | |
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Revision as of 09:43, 25 April 2011
Contents
MA 265 Chapter 3 Sections 3.1-3.2
By: Daniel Ford
What are determinants?
To understand determinants, you must first know about permutations.
Permutations
If D = {1, 2,....,n} a set of integers from 1 to n in ascending order, then a permutation is the rearrangement of an integer in D.
Example:
If D = {6, 7, 8, 9}, then 7689 would be a permutation of D. This corresponds to the function f: D→ D defined by
f(1) = 6
f(2) = 7
f(3) = 8
f(4) = 9
Then after permutation,
f(1) = 7
f(2) = 6
f(3) = 8
f(4) = 9
The reason that this is important to know is that the total number of permutations can be even or odd. So in the permutation 4321, where 4 precedes 3, 4 precedes 1, 4 prececedes 2, 3 precedes 1, and 3 precedes 2. Thus the total number of inversions in this premutation is 5, which would make 4321, odd. This determines what sign you put in front, so if it is odd, then you put a negative sign (-) in front; and if it is even, then you put a postive sign (+) in front.
