1. Dr. Shildneck Fall, 2015 SOLVING SYSTEMS OF EQUATIONS USING MATRICES

2. MULTIPLY THE FOLLOWING

3. NOW, TAKE THAT ANSWER AND SET EQUAL TO

4. SO, NOW WE HAVE…

5. WHICH THROUGH EQUIVALENT MATRICES GIVES US THE SYSTEM… NOW SOLVE THE SYSTEM…

6. YOU SHOULD HAVE GOTTEN THE ANSWER… (-2, 2)

7. NEXT, LET’S USE INVERSE MATRICES TO SOLVE THE MATRIX EQUATION…

8. FIRST, FIND THE INVERSE OF THE MULTIPLIER… Determinant =-2 – 15 =-17

9. NEXT, MULTIPLY BOTH SIDES BY THE INVERSE…

10. WHAT DO YOU NOTICE? So, what do you notice about our answers to the system and the matrix equation? What do you think Matrix X was?

11. GIVEN THAT THIS SYSTEM HAD THE SAME ANSWER AS THIS MATRIX EQUATION… WHAT CAN YOU CONCLUDE ABOUT HOW THE SYSTEM AND MATRIX EQUATION RELATE?

12. USE YOUR CONCLUSION TO WRITE THE FOLLOWIG SYSTEM AS A MATRIX EQUATION

13. DID YOU GET… COEFFICIENTS Matrix VARIABLES Matrix ARGUMENTS Matrix

14. TRY WRITING THIS SYSTEM AS A MATRIX EQUATION

15. SYSTEMS AND MATRICES The fact that we can write and NxN system as a Matrix Equation allows us to use Inverse Matrices to solve the Matrix Equation rather than multi- step algebraic manipulations. Furthermore, for systems bigger than 2x2, this process allows us to quickly solve the system in one step, rather than a page full of steps!

16. SYSTEMS AND MATRICES The process for solving any system of equations using matrices is as follows: 1.Write the system as a matrix equation AX=B, where A = the coefficient matrix, X = the variable matrix, and B = the argument matrix 2.Find the inverse of A and multiply. 3.The solution to the system is given by X = A -1 B

17. EXAMPLE 1 – USING A CALCULATOR Step 1: Write the Matrix Equation. Step 2: Enter Matrix A and Matrix B in the calculator. Step 3: Solve by multiplying A -1 B. Step 4: Write the solution as a set of coordinates.

18. EXAMPLE 2 – USING A CALCULATOR

19. ASSIGNMENT ASSIGNMENT #8 – Solving Systems of Equations with Matrices