(New page: Category:MA265Fall2010MominAl =MA 265 Chapter 3 Sections 3.1-3.5: A Review= =By: James Jacob= '''Section 3.1: Defining Determinants''' ---- Determinants are not as efficient as met...) |
|||
| Line 9: | Line 9: | ||
First there are '''permutations'''. If P = {1,2,.....,n} a set of integers from 1 to n in ascending order, then a permutation would be every rearrangement of an integer in P. | First there are '''permutations'''. If P = {1,2,.....,n} a set of integers from 1 to n in ascending order, then a permutation would be every rearrangement of an integer in P. | ||
| + | |||
| + | '''Example:''' | ||
For example if P = {4,5,6,7}, then 5467 would be a permutation of P. | For example if P = {4,5,6,7}, then 5467 would be a permutation of P. | ||
| Line 26: | Line 28: | ||
Permutations can have '''inversions''' if a larger integer comes before a smaller one in the set. If the total number of inversions is even, the permutation is '''even'''. If the total number of inversions is odd, then the permutation is '''odd'''. | Permutations can have '''inversions''' if a larger integer comes before a smaller one in the set. If the total number of inversions is even, the permutation is '''even'''. If the total number of inversions is odd, then the permutation is '''odd'''. | ||
| − | + | '''Example:''' | |
| + | |||
| + | If a permutation = 6543, 6 is larger and comes before 5, 4, and 3, 5 comes before 4 and 3, and 4 comes before 3 which totals to 6 inversions therefore the permutation is even. Now if the permutation = 6345, 6 comes before 3, 4, and 5 which totals to 3 inversions therefore the permutation is odd. If the number of elements ''n'' is greater than or equal to two in a set then there are ''n''!/2 even and ''n''!/2 odd functions. | ||
| + | |||
| + | |||
| + | Determinant is also written as '''det''' and is defined as the summation of all permutations of a set A. Each permutation is positive or negative depending on whether it is even or odd respectively. Each term of determinant set A is a product of ''n'' entries, so one entry from each row and one from each column. There are ''n''! terms in the sum. | ||
| + | |||
| + | '''Example:''' | ||
| − | + | If | |
| + | A = | ||
| + | <math>\begin{bmatrix} | ||
| + | a_{\text{11} & a_{\text{12}\\ | ||
| + | a_{\text{21} & a_{\text{22} \end{bmatrix}</math> | ||
| + | then | ||
| + | |||
| + | det(A) = <math> a_{\text{11}}a_{\text{22}} - a_{\text{12}}a_{\text{21}} </math> | ||
Revision as of 09:49, 16 December 2010
MA 265 Chapter 3 Sections 3.1-3.5: A Review
By: James Jacob
Section 3.1: Defining Determinants
Determinants are not as efficient as methods for solving systems like in Chapter 2. Determinants are also important in linear transformations when discussed in Chapter 6.
First there are permutations. If P = {1,2,.....,n} a set of integers from 1 to n in ascending order, then a permutation would be every rearrangement of an integer in P.
Example:
For example if P = {4,5,6,7}, then 5467 would be a permutation of P. First,
f(1) = 4 f(2) = 5 f(3) = 6 f(4) = 7
Then after permutation,
f(1) = 5 f(2) = 4 f(3) = 6 f(4) = 7
So any element in P can be in any position and each "new" set using the same elements is a permutation. The number of total permutations a set can have can be determined by the number n elements. The total number of permutations is equal to n! (n factorial).
Permutations can have inversions if a larger integer comes before a smaller one in the set. If the total number of inversions is even, the permutation is even. If the total number of inversions is odd, then the permutation is odd.
Example:
If a permutation = 6543, 6 is larger and comes before 5, 4, and 3, 5 comes before 4 and 3, and 4 comes before 3 which totals to 6 inversions therefore the permutation is even. Now if the permutation = 6345, 6 comes before 3, 4, and 5 which totals to 3 inversions therefore the permutation is odd. If the number of elements n is greater than or equal to two in a set then there are n!/2 even and n!/2 odd functions.
Determinant is also written as det and is defined as the summation of all permutations of a set A. Each permutation is positive or negative depending on whether it is even or odd respectively. Each term of determinant set A is a product of n entries, so one entry from each row and one from each column. There are n! terms in the sum.
Example:
If
A =
$ \begin{bmatrix} a_{\text{11} & a_{\text{12}\\ a_{\text{21} & a_{\text{22} \end{bmatrix} $
then
det(A) = $ a_{\text{11}}a_{\text{22}} - a_{\text{12}}a_{\text{21}} $
