1. 9/11 More systems of equations: RREF etc Augmented matrix of a system Use Row operations To get Row echelon form to determine how many solutions Proceed to Reduced row echelon form to get solutions (if any) x + 2y = 1 3x + 4y = 2 x y rhs 1 2 1 3 4 2

2. Common #2 Find RREF -5 -3/2 2 2 ½ 0

3. Mixture problems: give names to the various outputs and write the equations they must satisfy. Solve the system. Common #3 3 equations in three variables x y z rhs 1 1 1 234 1 -1 -1 0 25 35 56 8637

4. Consistent means: Common #4 Find b so that the system is consistent t – r = b -2t + r = 7 t – r = -4

5. Traffic flow problem: #

6. #7 find a so that the system does not have a unique soln -2z + 2x + v = -2 -2z +2v=-2 az + x + 2v =2

7. #8 Find k so that there is either no solution or an infinite Number of solutions to

8. Matrix operations: The idea traces back to mid 1800’s Replace a system of equations with a single matrix equation x + y = 1 x - y = 2

9. Stock portfolios Example of matrix multiplication Bob Tom and Jane stock holdings Stock IBM GE Toyota TI Stock Price Bob 20 30 14 2 IBM 30 Tom 10 40 80 10 Ge 100 Jane 50 21 10 24 Toy 75 TI 45 Portfolio matrix * price vector = portfolio value vector