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a1 a1 + a2 a3 or a1 + a2 a2 a3 b1 + b2 b2 b3 c1 + c2 c2 c3 d1 + d2 d2 d3 b1 b1 + b2 b3 c1 c1 + c2 c3 d1 d1 + d2 d3

Complex Number Vectors Complex numbers can also be represented as vectors This gives each complex number a unique magnitude and a unique direction The magnitude is the distance of the point a + jb from the origin 0 + j0 The direction is the angle of the vector, expressed counterclockwise from the positive realnumber axis This is shown in Fig 15-3 Absolute Value The absolute value of a complex number a + jb is the length, or magnitude, of its vector in the complex plane, measured from the origin (0,0) to the point (a,b) In the case of a pure real number a + j0, the absolute value is simply the real number itself, a, if a is positive If a is negative, then the absolute value of a + j0 is equal to a In the case of a pure imaginary number 0 + jb, the absolute value is equal to b, if b (a real number) is positive If b is negative, the absolute value of 0 + jb is equal to b If the number a + jb is neither pure real or pure imaginary, the absolute value must be found by using a formula First, square both a and b Then add them Finally, take the square root This is the length, c, of the vector a + jb The situation is illustrated in Fig 15-4 Problem 15-1 Find the absolute value of the complex number 22 j0 This is a pure real number Actually, it is the same as 22 + j0, because j0 = 0 Therefore, the absolute value of this complex number is ( 22) = 22 Problem 15-2 Find the absolute value of 0 j34 This is a pure imaginary number The value of b in this case is 34, because 0 j34 = 0 + j( 34) Therefore, the absolute value is ( 34) = 34.

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712 For the linear program max x1 2x3 2x2 x3 1 x1 , x2 , x3 0 prove that the solution (x1 , x2 , x3 ) = (3/2, 1/2, 0) is optimal 713 Matching pennies In this simple two-player game, the players (call them R and C) each choose an outcome, heads or tails If both outcomes are equal, C gives a dollar to R; if the outcomes are different, R gives a dollar to C (a) Represent the payoffs by a 2 2 matrix x1 x 2 1

Note that the replaced row must be one of the two involved in the sum In this example, we aren t allowed to replace the third row with the sum of the first and second rows

The final goal The matrix morphing game, like any sensible game, has an ultimate objective Our goal is to get a matrix representing a three-by-three linear system into this form:

Problem 15-3 Find the absolute value of 3 j 4 In this number, a = 3 and b = 4 Squaring both of these, and adding the results, gives us 32 + ( 4)2 = 9 + 16 = 25 The square root of 25 is 5 Therefore, the absolute value of this complex number is 5

714 The pizza business in Little Town is split between two rivals, Tony and Joey They are each investigating strategies to steal business away from the other Joey is considering either lowering prices or cutting bigger slices Tony is looking into starting up a line of gourmet pizzas, or offering outdoor seating, or giving free sodas at lunchtime The effects of these various strategies are summarized in the following payoff matrix (entries are dozens of pizzas, Joey s gain and Tony s loss) T ONY Gourmet Seating Free soda +2 0 3 1 2 +1

where x, y, and z are real numbers This is called the unit diagonal form Now imagine that we start with a three-by-three linear system, make it into a matrix, and then play the matrix morphing game until we get the unit diagonal form Do you suspect that the values x, y, and z, which appear in the far right column, will represent the solution to the linear system If so, you are right, provided the system is consistent (has a unique solution)

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