Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: [email protected]http://www.math.iastate.edu/hentzel/ class.307.ICN Text: Linear Algebra With Applications, Second Edition Otto Bretscher
26
Embed
Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Text: Linear Algebra With Applications, Second Edition Otto Bretscher
Wednesday, April 2 Chapter 6.1 Page 250 Problems 8,17,44,48Main Idea: There is a magical number associated to a matrix called the determinant. Key Words: Determinants, Elementary Row OperationsDet[A]=SUM sgn(p) a1p(1) a2p(2) ... a np(n) all p Goal: Learn the definition of a determinant. how the elementary row operations affect the determinant. det (A B) = det (A) det (B)
Today we do the following.
1. Give definition of the determinant of A n x n
2. Show how the elementary row operations affect the determinant.
3. Show that det [ A B] = det [A] det[ B].
The definition in English. Given A n x n
(a) Pick n elements of A no two of which are in the same row or column.
(b) Multiply then together.(c) Calculate the appropriate sign. (d) Do this for all possible choices of n
elements. (e) The sum of these terms is the determinant
of A.
The definition using summation notation.Det[A] = SUM sgn(p) a1p(1) a2p(2) ... a np(n)
all p The n elements are from n different rows, the one in the first row is a 1 p(1) ; the one in the second row is a 2 p(2) ; etc. Sgn(p) is the calculated sign. Sgn(p) depends on the second coordinates of the chosen elements.
The sign of the term is computed by counting the number of times that a larger number precedes a smaller number in the secondsubscripts p(1) p(2) ... p(n).
If that number is even the sgn is +1, if that number is odd, then the sgn is -1.
Example: a14 a23 a32 a41
look at 4 3 2 1 there are six instances where a larger precedes a smaller. The sgn is +1.
Example: a11 a23 a32 a44
look at 1 3 2 4 there is one instance where a larger number precedes a smaller. The sgn is -1.
Now we look at how the elementary row operations affect the determinant.
(1) Switching two rows changes the sign of the determinant.
(2) Multiply a row by a scalar c multiplies the determinant by c.
(3) Adding a multiple of one row to another does not change the determinant.
Elementary Row Operation (1).We first show that the statement is true if we switch two adjacent rows. When the rows are switched, the sign is computed with the positions p(i) and p(i+1) interchanged. This interchange increases the number of times a larger precedes a smaller when p( i ) < p(i+1).This interchange decreases the number of times a larger precedes a smaller When p( i ) > p(i+1). Since the sign changes for each term, the sign of the determinant is changed.
If the rows switched are not adjacent, we do a series of adjacent switches which have the end result we want. Ri ------- xx | xx | r rows xx | ------| RjWe switch the Ri with each of the intermediate rows,Switch Ri and Rj, and then switch Rj with the intermediate rows. The final result has only Ri and Rj switched. In the process we made 2r+1 switches. The determinant has changed sign 2r+1 times and now is opposite what it originally was.
Elementary Row Operation (2) Multiplying a row by a scalar c multiplies the determinant by c. | ---R1 --- | | --- R1 ----| | ---R2 --- | | --- R2 ----| | --- --- | | --- ----| Det | c Ri | = c Det | Ri | | --- --- | | --- ----| | --- --- | | --- ----| | ---Rn --- | | --- Rn ----|
If we multiply the elements of a row by c, there appearsexactly one c in each of the terms. We can factor the c out and place it before the summation.
If A’ is A with row i multiplied by c then
Det[ A’ ] =SUM sgn(p) a 1 p(1) ... (c a i p(i) ).... a n p(n).
all p = c SUM sgn(p) a 1 p(1) ... a i p(i) .... a n p(n).
all p = c Det[ A ].
Elementary Row Operation (3). Adding a multiple of one row to another does not change the determinant. | --- R1 --- | | --- R1 ----| | --- R1 ----| | Ri + c Rj | | --- Ri ----| | --c Rj ----| | --- --- | | --- ----| | --- ----| Det | --- --- | = | --- ----| + | --- ----| | --- Rj --- | | --- Rj ----| | --- Rj ----| | --- --- | | --- ----| | --- ----| | --- Rn --- | | --- Rn ----| | --- Rn ----|
This is true because each term in the summation has one entry from Ri + c Rj.
The term can be expanded into two pieces. Put the part from Ri into the first and the part from c Rj into the second.