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MATH 1502 (10)

# PracticeProblems3Fall2012.pdf

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School
Georgia Institute of Technology
Department
Mathematics
Course
MATH 1502
Professor
Blekherman
Semester
Fall

Description
Practice Problem for Midterm #3, Math 1502 1. Find and simplify the following determinants: ▯ ▯ ▯ ▯ ▯ ▯ ▯ -1 0 5 ▯ ▯ ▯1 b ▯ ▯a 1 2 ▯ a)▯ 1 -2 -1 ▯ b)▯ ▯ c)▯a 2 ▯1 ▯ ▯ 3 -1 2 ▯ x y ▯a ▯1 2 ▯ ▯ ▯ ▯-1 3 3 4 ▯ ▯ ▯1 ▯ ▯ 0 5 ▯ ▯ ▯ d)▯ 1 ▯2 ▯ ▯ ▯1 ▯ e) ▯0 -2 1 3 ▯ ▯ ▯ ▯0 0 3 4 ▯ 3 ▯1 2 ▯ ▯ ▯0 0 0 3 ▯ 2. Find the characteristic polynomial, eigenvalues and eigenvectors of the following matrices: ▯ ▯ ▯ ▯ ▯ ▯ -2 4 2 1 4 1 a) 9 -7 b) 12 1 c) 3 2 0 1 -1 3 3 4 0 1 0 1 B 0 -2 1 3 C 1 0 1 1 1 1 d)@ A e)@ 0 1 ▯1 A f)@ 2 2 2 A 0 0 3 4 5 ▯1 2 3 3 3 0 0 0 3 3. Find the inverses of the following matrices, if possible: 0 1 0 1 ▯ ▯ ▯ ▯ 1 1 5 2 1 -1 1 3 1 4 @ A @ A a) 2 -6 b) 2 8 c) 3 2 1 d) 1 2 1 2 1 -2 -3 0 -2 4. Find the basis of the kernel and the column space of the following matrices and determine their rank. 0 1 ▯ ▯ 0 1 4 1 0 1 3 1 1 3 2 -3 1 B -1 2 C a) 2 6 1 -2 4 b) @ 2 -4 3 -1 A c)@ 2 1 A -2 3 -5 2 -5 6 5. Find the dimension and a basis of the following subspaces: 0 1 1 0 4 1 0 3 1 a) The span of 6 A , -1 A and @ -4 A. -2 3 4 0 1 x1 B C b) All vectors x2C in R with 1 + 2 + x3+ x4= 0. (Hint: Think of this as a kernel of some @ x3A x4 matrix A). 1 0 1 x1 3 c) All vectorsx2 A in R with 1 + 2 ▯ 33 = 0 and 1 ▯ 2 + 23 = 0 (Hint: Think of this as x3 a kernel of some matrix A). 6. Diagonalize the following matrices, if possible: 0 1 0 1 0 1 ▯ ▯ 1 4 1 2 0 1 2 0 0 0 1 -2 B 0 1 2 0 C a) 2 6 b)@ 0 -4 3 A c)@ 0 1 0 A d)@ 0 0 -5 0 A 0 0 2 1 0 2 0 0 0 -3 7. True or False. No partial credit. (a) A singular matrix always has deter
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