Math 1314 College Algebra 7.6 Solving Systems with Gaussian Elimination (Gauss-Jordan Elimination)
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1 Math 1314 College Algebra 7.6 Solving Systems with Gaussian Elimination (Gauss-Jordan Elimination) A matrix is an ordered rectangular array of numbers. Size is m x n Denoted by capital letters. Entries are written a i j for any matrix A. We can write a system of equations as an augmented matrix (sometimes called a system matrix): 3x + 5y + 2z = 22 6x 4y + z = 28 2x + 3y + 5z = 9 A matrix is in Row Echelon form when: 1. First nonzero entry in a row is 1. (called lead unit or leading 1) s are in a stairstep (1 below and to right of previous 1). 3. Any row consisting of all zeroes is at the bottom (i.e. last row). Ex: Are these matrices in row echelon form? (a) [ (b) ] (c) [ ] (d) [ ] (e) A unit column has one 1 and all other entries are 0. Row-reduced Echelon form: Has same conditions as row echelon form AND every column with a leading 1 must be a unit column. (RREF)
2 When solving a system of equations, we would like our final augmented matrix to be in row-reduced echelon form so we can easily read-off the solutions. For a system with: 3 equations and 3 variables: 2 equations and 2 variables: GOAL: a [ ] b 1 0 a GOAL: 0 1 b c Row Operations: 1. Interchange any two rows. R i R j 2. Replace any row with a nonzero constant multiple of itself. cr i R i 3. Replace any row with the sum of the nonzero multiple of another row and itself. R i + cr j R j Notice there are no equal signs. These are not equal. Process: Go column by column (left to right) and make unit columns. Start with the 1,1 entry and change that entry to 1. Two options for the 1,1 entry: Row Op 1 (interchange row 1 with another row that has a 1 in the first column) or Row Op 2 ( 1 # to change Next, get zeroes in that column to make a unit column. ) (Row to change) (Row to change) Use Row Op 3: (-# to change)*(row with leading 1) + (Row to change) Row to change Move over to the next column to the right. Get the leading 1 in the appropriate entry (in the stairstep). NOTE: Cannot use Row Op 1 for any entry other than the 1,1. Need to use Row Op 2. Get 0 s in that column. Use Row Op 3. Continue until all columns are unit columns (one 1 and all other entries are 0). NOTE: This process works every time if you are nervous. You may see other ways to get the matrix into row-reduced echelon form. That is fine, as long as the row operations are done correctly. We are using the multiplicative inverse of the number we are trying to change in Row Op 2. We are using the additive inverse of the number we are trying to change in Row Op 3. Since we created a leading 1 in that column already, it is convenient to use that to get our 0. 2
3 3x 5y = 19 x + 2y = 1 If a 11 ( entry is not 1, interchange ) two rows or use RowOp2 1. (Row to change) (Row to change) # to change to 1 Change any other entries in column to 0 with RowOp3. ( # to change to 0)(Row with the leading 1) + (Row to change). (Row to change) Have you successfully pivoted on the a 11 entry? Do you have a unit column? Move ( over a column and ) make the leading 1. Use RowOp2 1. (Row to change) (Row to change) # to change to 1 Change any other entries in column to 0 with RowOp3. ( # to change to 0)(Row with the leading 1) + (Row to change). (Row to change) Have you successfully pivoted on the leading 1? Do you have a unit column? Continue with the other columns until all are unit columns or the matrix is in row-reduced form (or row-reduced echelon form). Read off solutions. 3
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