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.

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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

Common #2 Find RREF -5 -3/2 2 2 ½ 0

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

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=-2 az + x + 2v =2

#8 Find k so that there is either no solution or an infinite Number 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 = 1 x - y = 2

Stock portfolios Example of matrix multiplication Bob Tom and Jane stock holdings Stock IBM GE Toyota TI Stock Price Bob IBM 30 Tom Ge 100 Jane Toy 75 TI 45 Portfolio matrix * price vector = portfolio value vector