PROBLEM 1 [3 points] a. Alice and Bob have N quantum computers. They divide them

Question

PROBLEM 1 [3 points] a. Alice and Bob have N quantum computers. They divide them up in such a way that Alice has three times plus two as many as Bob, while Bob has four less than Alice. Write down two linear equations that describe this information and solve them. How many quantum computers N do they have in total, and how many each, na,nb? b. Find the same solution graphically as the intersection of two lines in the (na, nb) plane, c. Give a general proof of the property that n-(Ca, Cb) 1s the (unnormalized) normal vector d. Bring the equations of part(a) into the 2x2 matrix form ã A- where each line represents one of the two equations in part (a) of the line Cana-cbn,-1 in the {na,nb} plane ab with r= (na, n.). Find the determinant of matrix A. What is the rank of A? e. Find the inverse matrix A-1. check explicitly that your matrix obeys, ?? f. Check explicitly that your inverse matrix reproduces correctly eq. (6.13) in Boas, which ir 2D reads ab det(A) -Ava A. g. Check that Cramer's rule, eq.(3.8) on page 93 in Boas, is equivalent to A-'ä after inserting this formula.

Explanation / Answer

a. Equations are: x=3y=2 and y=x-4 ,which on solving gives x=5 and y=1 .So,they have total 6 quantum computers,where Alice has 5 and Bob has 1.    d. [-2]= [ -1 3] [x]    -4 -1 1 y    DETERMINANT OF MATRIX A=2 ,RANK OF A= 1

e. 1/2 * [-1    -3] -1    -1

-> -> ~ r = A^-1 a

[x] = [-1/2    -3/2]    [-2]    y    1/2 1/2 -4

f.    A^-1 = 1/2 * [-1 -3] 1 1

g. There is no pic available of page 93

b.Need to put up a image of graph

c. Its the normal property.   

  

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