1.2 Matrices Vectors and Gauss – Jordan elimination (This is such an interesting topic that it was made into a movie)

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Presentation transcript:

1.2 Matrices Vectors and Gauss – Jordan elimination (This is such an interesting topic that it was made into a movie)

Matrix Notations A website that has programs that will do most operations in this course (an online calculator for matrices) http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi

Write the system as a matrix Write the following matrix in reduced row echelon form (example taken from p. 12 of text) 2x + 8y + 4z = 2 2x + 5y + z = 5 4x + 10y – z = 1

Solution

Solution slide 2

What are legal steps for reducing a matrix? You are allowed to interchange rows. You are allowed to multiply a row by a constant. You are allowed to add two rows together. (This implies that you can multiply a row by a constant then add it to another row.)

What is the order that a matrix should be simplified? Step 1: Get a 1 in the upper left hand corner. Step 2: Obtain 0’s for the rest of the first column Step 3: Get a 1 on the main diagonal in the next column. Step 4: Get zeros below the one obtained in step 3 Step 5: return to step 3 and repeat steps 3 and 4 until there are 1s on the main diagonal and zeros below it. Step 6: start on the right most column and get zeros above the main diagonal. Repeat this for all diagonals from right to left.

Reduced Row-Echelon (rref) Form A matrix is in reduced-row echelon form if it satisfies all of the following conditions: If a row has nonzero entries, then the first nonzero entry is 1 called the leading 1 in this row. If a column contains a leading 1, then all other entries in that column are zero If a row contains a leading 1, then each row above contains a leading 1 further to the left

Which matrices are in Reduced Row Echelon form Which matrices are in Reduced Row Echelon form? If a matrix is not in rref then what changes would be needed to change it to that form?

Write the given matrix in reduced row echelon form using a TI-89 Calculator 2nd 5 (math) 4 (matrix) 4(rref) - rref stands for reduced row echelon form rref([1,3;2,5])

Homework p. 18 1-19 odd,18, 20, 21, 22, 27 Q: How does a mathematician induce good behavior in her children? A: `I've told you n times, I've told you n+1 times...'