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# The top universities in the U.S. are building up a data transmission network to share their resources

INSTRUCTIONS TO CANDIDATES

Instructions

Submit LastName_FirstName_midterm.m to Blackboard.

If a MATLAB function is specified in the prompt, the function must be applied to the code to receive a full scroe.

Non-runnable scripts will receive a zero.

Use sections to separate code for each problem. The script you submit shall have 2 sections (no more, no less).

Do not use user-input in the exam. Only output requested items.

Please finish all the problems on your own. Sharing/Receiving scripts from others is condered as giving/using unauthorized help.

Exam Problem 1 2

Points (80) (75)

Problem 1 The top universities in the U.S. are building up a data transmission network to share their resources. The number shown in each branch are in the unit of GB/sec.

(a) Find all the branch capacities xi for i = 1, 2, . . . , 6 in the network. All the flows shall be combined into one vector x ∈ R6 where xi is the i-th element in the vector. Use the display() function to output x. (25 pt)

Hint: Lay out the equations on the paper and input the matrix form into MATLAB.

More hint: What is the matrix form of a linear system? Is x a column vector or a row vector?

(b) Continuing from part (a), remove the equation of node #5 (U.C. Berkley) from the linear system, and then rewrite the system to the form Cx = d where C ∈ R6×6 and d ∈ R6×1. Output the determinant of C in your code. (30 pts)

Hint: the augmented matrix form of Cx = d shall look like Eq. (1).

eqn of node1    −

eqn of node2    −

eqn of node3    − . (1)

eqn of node4    −

eqn of node6    −

eqn of node7    −

(c) Continuing from part (b), apply the inv() function to solve for the vector x. Output x in your code. (25 pts) Hint : Why can’t we use inv() in part (a)?

Problem 2 Given the following matrices

  1 0 0 

  cos(60◦) −sin(60◦) 0 

 0   0 

x1 =   1.5   , x2 =   2   .

(a) Let y1 = M1x1 and y2 = M1M2x2, and y3 = y1 • y2 (y3 is the inner product of y1 and y2). Use the

transpose() function to find y3. Output y3. (25 pts)

(b) Find M1−1  and M2−1  and assign the values to the variables named m1_inv and m2_inv, respectively.  Find

M3−1  in terms of m1_inv and m2_inv where

M3 = M 2M1 I3×3.

Output M3−1. (25 pts)

Hint: If you do ans = inv(M2*M2*M1*eye(3)), a zero will be recorded.

(c) Find det(M1) and det(M2) and assign the values to the variables named m1_det and m2_det, respectively. Find and output the determinant of m3 in terms of m1_det and m2_det. (25 pts)

Hint: sind() and cosd()

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