1. Warm-Up

2. Reduced Row Echelon Form (RREF)

3. Learning Targets  Possible solutions for a system  The differences between RREF and Inverse Multiplication  Using Reduced Row Echelon Form to solve systems

4. Quick Recap  In order to setup a matrix our data must be:  Setup in the Standard Form  All similar variables must be in the same order  Any missing variables in the equations must be represented with a zero  Our matrix system using inverse multiplication has three matrices:  Coefficient, Variable and Constant

5. Recap Cont.  Matrices cannot divide one another  Multiplying the coefficient matrix by its inverse isolates the variable matrix  Multiplying the constant matrix by our inverse as well will solve for the variables

6. How many types of solutions can we have?  Pause and Ponder, SILENTLY!  One Solution  No Solution  Infinite Solutions

7. Example

9. Problems with Inverse Multiplication  Inverse multiplication only produces a solution when there is only one solution.  If we have no solution or infinite solutions then we will get an ERROR  So how do we know if it is infinite solutions or no solution?

10. RREF  Reduced Row Echelon Form  This form allows us to consolidate the coefficient and constant matrices into one matrix  We can then perform row operations that will clearly state the exact answer

11. How to set it up: CoefficientVariable Constant

12. How to set it up: Coefficient Constant ***Notice there is no more variable matrix and we add the constant as an additional column.

13. Now to perform the math…  In our calculators we can go under the matrix screen and select RREF  We can then choose the matrix to perform this operation on.

14. Solution  The solution will be in a matrix that is the same dimensions.  We can then read the results as variables and their solutions. X-Variable Y-Variable Z-Variable

15. No Solution: Enter as a 3X4 matrix Last row: 0 0 0 1 No Solution.

16. Infinite Solutions: Enter as a 3X4 matrix Last row: 0 0 Infinitely Solutions

17. You Try using RREF:

18. Website to Visualize the Solutions  http://www.cpm.org/flash/technology/3dsystems.swf http://www.cpm.org/flash/technology/3dsystems.swf

19. For the weekend…  Worksheet