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please explain the answers clearly and show every steps.(I will study to final exam by your work) Just start from the problem 3(Including problem 3)

Assignment Description

University at Buffalo

EAS 230 – Fall 2016 – Final Examination

DURATION: 2 hours


SCORE:                /100 (max 110)




1.      Do not open the exam booklet until you are told to do so. 

2.      This exam booklet consists of 11 total pages: this cover page, 1 page of Reference Materials and 8 problems. 

3.      The back of each problem page was left empty so you can write on it, if you wish. The cover page and the reference material page can be considered scrap papers and will not be graded.

4.      You must place your UB Card (or other form of valid ID) on the desk in front of you during the exam. 

5.      No electronic devices, including calculators, and no outside “reference sheets” are allowed. 

6.      No bathroom breaks except for students having health problems. 

7.      Your electronic devices (cell phone, tablet, etc.) should be placed in your backpack which must be zipped/closed and placed on the floor at the classroom blackboard.

8.      Writing on body, clothes, etc. is considered cheating and may result in a grade of zero on a problemor a grade of zero for the entire exam.

9.      Be sure to write your name and person number on each page. 

10.  Be sure you have all pages of the exam booklet and they are all stapled together. 

11.  To get full credit, you must show all work in this booklet (you may write on the back of each sheet). 

12.  You must stop writing when told to do so or you will receive a grade of zero on the problemor a grade of zero for the entire exam. 

13.  If you have a question, please remain seated, raise your hand and wait for a proctor to come to you. 

14.  You must turn-in everything: your exam booklet, your “bubble-sheet” (if any), and your scrap paper (however, the scrap paper will not be graded). 

15.  When you turn-in your exam papers, you must fill-out the sign-in sheet with your name, person number, lab section and signature.


EAS 230 – Fall 2016 – Final Exam                                                                                     Page 

Reference Materials


Some of the MATLAB built-in functions and their descriptions:

MATLAB Functions



Returns the matrix dimensions [#rows #columns]


Returns the length of a vector (number of elements)


Returns the determinant of a square matrix


Returns the inverse of a square matrix


Returns a matrix where all entries are ones


Returns a matrix where all entries are zeros


Returns the identity matrix


Problem 1 (10 pts)

Write the output of the following commands when executed sequentially in MATLAB command window, if the command is not suppressed with (;) the output should be displayed.

- If variables are assigned names write the output as “ variable name = the value you calculate” - If variables are not assigned names write the output as “ ans = the value you calculate “ - if MATLAB returns error, write “error” and the reason of that error.


>> A = [2:-2:-2; 1:3] 



>>[m , n] = size(A)



>> Asq = A.^2



>> AAM = A * A



>> AAS = A + Asq


>> det(A)



>> inv(A)



>> A(3,:) = 2:2:6



>> det(A)


Problem 2 (20 pts)

a.       (10 points) Complete the following script to produce the adjacent figure 

 close all x = -----:pi/20:-----;

-----(x, -----, -----, cos(x),'-.'); -----('sin(x)','cos(x)'); axis([-----------------]);

-----('angle in radians')


-----('Plot of sin(x) and cos(x) from -2pi to 2pi')

----- on



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b.      (10 points) Write the output of the following Matlab scripts as it appears in the command window? 


i.              (5 points)

 clear a = 1; while a < 8     if a == 3         a = a * 2;         continue;


    fprintf('value of a: %d ', a);     a = a + 1; end










ii.            (5 points)

 clear A = [5 3 7 4]; iter = 0; M = A(1); for ii = 1:length(A)     iter = iter+1;     if A(ii) > M         M = A(ii);     end

    disp(['iter:' num2str(iter) '  M = ' num2str(M)]) end

Problem 3 (15


Given the function get_product shown below, write the output of the commands shown after the function end as they are displayed in the command window.


function PRODUCT = get_product(MATRIX_A,MATRIX_B)

[RA,CA] = size(MATRIX_A); [RB,CB] = size(MATRIX_B); if (RA ~= CA) || (RB ~= CB)

    error('Either one of the matrices entered or both are not square') elseif (RA ~= RB)

    error('Matrices dimensions are not the same')

elseif det(MATRIX_A) >= det(MATRIX_B)     PRODUCT = MATRIX_A * MATRIX_B; else


end % end of the function




>> get_product ([1 1 ; 0 1] , [1 -1 ; 1 0])



>> get_product ([1 1]' , [-1 -1])



>> get_product (ones(2) ,[1 -1 ; 1 0])



>> get_product (eye(2) , [0 1 ; 1 0])



>> get_product (eye(3) , zeros(2)])



4 (10

Use the proper elimination matrix/matrices to determine the L and U decomposition of the

2            1 matrix  𝐴 = [       ],

                              6    9

a.       (4 points) Without partial pivoting (permutation),

b.       (6 points) With partial pivoting (permutation),


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5  (10

Use Cramer’s rule to solve the following system of equations:


3𝑥1 + 𝑥2 = 3

2𝑥1 − 𝑥2 = 7


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6  (15

For the following system of equations


𝑥1 + 2𝑥2 = 4

2𝑥1 − 𝑥3 = 1

2𝑥2 + 𝑥3 = 5


a.    (2 pt)  Rewrite the system of equations in 𝐴𝐱 = 𝐛 form.

b.   (4 pts)  find rank(𝐴) and rank(𝐴|𝐛) by using Gaussian elimination to reduce 𝐴 and (𝐴|𝐛) to row echelon form (REF). 

c.    (2 pt)  With this information, determine whether the system of equations is consistent or inconsistent

d.   (4 pts)  If the system is consistent, find the solution (i.e. determine 𝑥1, 𝑥2 and 𝑥3) by performing back-substitution.

e.    (3 pts)  If each row of matrix 𝐴 can be considered as a row vector in a set of vectors 𝑆 =  {𝐑1, 𝐑2, 𝐑3}. From your solution of part b, is this set of vectors linearly dependent or independent and what is its span?


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7 (10

If a system of equations can be written in the form 𝐿 𝑈 𝐱 = 𝐛 as







0    1

0] [0

1    0




0           𝑥1                     4

−1] [𝑥2] = [1]

0.5     𝑥3                       5


where L and U are the Lower and Upper triangular matrices, respectively.

a.       (8 pts) solve this system of equations using forward and back substitution.

b.       (2 pts) Rewrite this system in the format 𝐴 𝐱 = 𝐛



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8 (20 pts)

a.       (6 points) Find the eigenvalues and eigenvectors of the matrix [1          4]

                                                                                                                                                                2    3

b.      (14 points) The following homogeneous linear system of differential equations is required to be solved by diagonalization:

                  𝐱= [−1      −3] 𝐱 with the initial conditions  [𝑥1(0)] = [0]  where 𝐱= 𝑑 𝐱 .

                                 2       4                                                   𝑥2(0)        1                        𝑑𝑡

If the eigenvalues of this system are 1 and 2 and the corresponding eigenvectors are 

                                 −3                    −1

                   𝐱𝟏 = [       ] and 𝐱𝟐 = [      ]

                                  2                        1

a.       (2 pts) Find the matrix of the eigenvectors 𝑋,

b.       (3 pts) Show that the matrix 𝑋 is not singular and find its inverse 𝑋−1 ,

c.        (1 pt) Find the diagonal matrix 𝛬,

d.       (4 pts) Use the substitution 𝐱 = 𝑋 𝐲 and find the general solution of the system, i.e., find


𝐱 = [𝑥2] in terms of 𝑡.

e.       (4 pts) Use the initial conditions to determine the arbitrary constants and write the solution in its final form.


Hasn’t been given yet          


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