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# quiz3! - 18.06 Professor Strang/Ingerman Quiz 3 December 6...

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18.06 Professor Strang/Ingerman Quiz 3 December 6, 2002 Your name is: Please circle your recitation: 1) M2 2-131 P.-O. Persson 2-088 2-1194 persson 2) M2 2-132 I. Pavlovsky 2-487 3-4083 igorvp 3) M3 2-131 I. Pavlovsky 2-487 3-4083 igorvp 4) T10 2-132 W. Luo 2-492 3-4093 luowei 5) T10 2-131 C. Boulet 2-333 3-7826 cilanne 6) T11 2-131 C. Boulet 2-333 3-7826 cilanne 7) T11 2-132 X. Wang 2-244 8-8164 xwang 8) T12 2-132 P. Clifford 2-489 3-4086 peter 9) T1 2-132 X. Wang 2-244 8-8164 xwang 10) T1 2-131 P. Clifford 2-489 3-4086 peter 11) T2 2-132 X. Wang 2-244 8-8164 xwang

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1 (36 pts.) (a) What are the eigenvalues of the 5 by 5 matrix A = ones (5) with all entries a ij = 1? Please look at A , not at det( A - λI ). (b) Solve this differential equation to find u ( t ): d u dt = A u starting from u (0) = (0 , 1 , 1 , 1 , 2) . First split u (0) into two eigenvectors of A . (c) Using part (a), what are the eigenvalues and trace and determinant of the matrix B = same as A except zeros on the diagonal. 2
3

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2 (20 pts.) (a) If A is similar to B show that e A is similar to e B . First define “similar” and e A !!
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