9/11 More systems of equations: RREF etc

Augmented matrix of a systemUse Row operationsTo get Row echelon form to determine how many solutionsProceed to Reduced row echelon form to get solutions (if any)

x + 2y = 13x + 4y = 2

x y rhs1 2 13 4 2

Common #2 Find RREF

-5 -3/2 2-5 -3/2 22 ½ 0

Mixture problems: give names to the various outputs andwrite 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

Consistent means:

Common #4 Find b so that the system is consistent

t – r = b-2t + r = 7 t – r = -4

Traffic flow problem: #

#7 find a so that the system does not have a unique soln

-2z + 2x + v = -2-2z +2v=-2az + x + 2v =2

#8 Find k so that there is either no solution or an infiniteNumber of solutions to

Matrix operations: The idea traces back to mid 1800’s

Replace a system of equations with a single matrix equation

x + y = 1x - y = 2

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 30Tom 10 40 80 10 Ge 100Jane 50 21 10 24 Toy 75 TI 45

Portfolio matrix * price vector = portfolio value vector