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WUSTL ESE 318 - Exam 2 Solutions and Grading

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ESE 318-02 Exam 2, Mar. 10, 2016 NAME:____68 exams of 69 enrolled ______ 1. (10 points) Find the determinants of the following matrices. ⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛−−−−−−=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛−−=425144829472326191067200013300237521BA A: Switch rows 1 and 3, then switch rows 2 and 4 (2 minuses cancel). Upper triangular remains: |A| = 2(-1)(3)(½) = -3. B: Columns 1 and 3 are multiples of each other, so |B| = 0. Grading: 47 students got full credit. 5 points each, all or nothing, except if you got 3 (instead of -3, a sign error in the answer) for |A|, I took off only 2 points. Comment: I said in class that I would never make you compute the determinant of anything bigger than a 3 X 3 the hard way; you would use properties to do it. I think I even said that in answer to a student question on Tuesday, March 8. 2. (10 points) For the matrix shown, find (a) its adjoint, and (b) its inverse. ⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=510310002A ( )( )⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−=⇒⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−==−=−212123252110000220610000241220610000226021000024352AAadjCA 52 papers received full credit. Everyone who didn’t simply skip part (b) got part (b) correct, given their part (a). 7 pts for adjoint, 3 for inverse. • Mis-calculated coefficient due to sloppiness: -2 pts. • Did not handle signs correctly (the 6 and the 2): -4 pts. • Wrongly calculated minor determinants: -4 pts. • Did not transpose (1 paper): -4 pts. • Multiple errors as above, yielding totally wrong adjoint: -7 pts. • Skipped part (b): -3 pts.ESE 318-02 Exam 2, Mar. 10, 2016 3. (10 points) For the matrix A below, find (a) all eigenvalues and (b) all corresponding eigenvectors. ⎟⎟⎠⎞⎜⎜⎝⎛−−=4323A ( )( )( )( )⎟⎟⎠⎞⎜⎜⎝⎛−==+⎟⎟⎠⎞⎜⎜⎝⎛−−=⎟⎟⎠⎞⎜⎜⎝⎛−==+⎟⎟⎠⎞⎜⎜⎝⎛−−−==−==−+=−+=+++−−=+−−−=−−−−12026321:231031326:3230236643126434323221212112122KkkKkkλλλλλλλλλλλλλλλ Or any scalar multiples of those eigenvectors. 61 papers got full credit. • “Sloppy” calculation: -2 pts. each. • Eigenvectors “upside-down”: -4 pts. • Bad eigenvalues & everything else: 0 pts.ESE 318-02 Exam 2, Mar. 10, 2016 4. (20 points) The characteristic equation of A has already been factored for you. (a) Find as many linearly independent eigenvectors as possible. (b) Can A be diagonalized? If so, find P and D such that P-1AP = D, where D is diagonal. ( )( )221111111113−−−=−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−=λλλIAA ⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=+=⇒⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−−−==⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛===⇒⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−→⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−=101011000111111111:2111000110101110000011101112:1323213211321KKkkkKkkkλλλ 3 LI eigenvectors for 3 X 3, so it can be diagonalized. P =1 1 11 1 01 0 1⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟D =1 0 00 2 00 0 2⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟ 35 papers received full credit. 13 points for part (a) and 7 for part (b). There were many ways to get errors. It was difficult to categorize them. But here’s a rough accounting: • “Minor” calculation mistakes: -2 to -3 pts. • Serious, or conceptual, errors in part (a), such as not understanding free parameters, number of LI eigenvectors, the “1-0-0-1” trick, etc.; wrong number of LI eigenvectors; eigenvector of all 0’s; getting wrong eigenvalues when it was already factored!; row reduction impossible to follow, or nonsense. o 3 correct eigenvectors, but too many “LI”: -4 pts. o Only 2 correct eigenvectors: -5 pts. o Only 1 correct eigenvector: -8 pts. o 0 correct eigenvectors: -10 pts. • Serious, or conceptual, errors in part (b), such as P and/or D not even the right dimension; not understanding criteria for diagonalizability; no idea where D came from; getting P and D backwards. o Said “not diagonalizable”, but gave correct P & D: - 3 pts. o Said “not diagonalizable” because of incorrect K’s: -3 pts. o P & D mixed up: -3 pts. o Bad P, bad D, or both bad (given your part (a)): -3 to -5 pts. o Said “not diagonalizable” (for no reason, or wrong reason) and gave no P & D: -7 pts. o Part (b) completely missing: -7 pts.ESE 318-02 Exam 2, Mar. 10, 2016 5. (10 points) One complex eigenvalue and its corresponding eigenvector have been found for the given system in X. Write the general solution for X. (Answer must be in all real terms.) ⎟⎟⎠⎞⎜⎜⎝⎛−==⎟⎟⎠⎞⎜⎜⎝⎛−−=ʹ3453455411iKiXXλ ( ) ( ) ( ) ( )⎟⎟⎠⎞⎜⎜⎝⎛+−+⎟⎟⎠⎞⎜⎜⎝⎛+=+=⎟⎟⎠⎞⎜⎜⎝⎛+−=⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛+=⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−−⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛−==⎟⎟⎠⎞⎜⎜⎝⎛======tttctttcXcXcXtttettXtttettXKBKB3sin43cos33sin53sin33cos43cos53sin43cos33sin53sin453cos303sin33cos43cos53sin303cos4530Im45Re3Im0Re2122110201121111λβλα 37 papers received full credit. Since this was essentially a plug&chug exercise, I was not generous with partial credit. If you didn’t even have the right form, you got 0 points. • “Sloppy” sign error: -1 pt. • Sign error on β or B2: -3 pts. • No general solution, or put together wrong: -4 pts. • Totally wrong form (like B’s not vectors, or in complex terms): 0 pts.ESE 318-02 Exam 2, Mar. 10, 2016 6. (20 points) Consider these two vectors: a = 0, 2, −1 b = −3, 0,1. (a) Find a vector that is parallel to a, and has the magnitude of b. (b) Find a unit vector perpendicular to both a and b. (c) Write an equation of a line, in symmetric form, that is perpendicular to both a and b and goes through the point P(2,-3,4). (d) Two lines, one parallel to a and the other parallel to b, go through the point P(2,-3,4). Write an equation of the plane containing those two lines. ( )2,22,01,2,022510105141,2,010109−=−=⇒±=±===+=−==++===ckkkkcbckaca ( )76737271,,6,3,2496,3,236946,3,26,3,2103120===++==−−=×××=ckjibababacb ( ) ( )bfromzyxc643322 −=+=− d( )2 x − 2( )+ 3 y + 3( )+ 6 z − 4( )= 0 from b( )or 2x + 3y + 6z = 19 31 papers received full credit. 5 points per part – mostly all-or-nothing. I


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